Airbnb is going with a more direct adjacency: Hotel bookings.
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KB: CPQ is a natural adjacency to what we do.
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"This is a true adjacency that complements compute virtualization," Casado said.
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As in much of Dylan's work, racial adjacency is a theme.
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Instead, he relies on his own instinct for adjacency and exploration.
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It's because of the legitimacy and the adjacency from the content.
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In its atmosphere, a discipline is a reminder of adjacency to eternity.
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It was things like measuring adjacency preference, interconnectivity, daylight, and views to the outside.
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The two countries have set up an "adjacency zone" 2km wide to separate them.
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But their adjacency to false conspiracy theories can make them seem more legitimate by association.
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The best that could be hoped for was the intimate adjacency of the facing page.
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I get how the anal adjacency of these glandular secretions makes castoreum repulsive to many people.
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Adjacency to import-export and Chinese manufacturing seemed to be a common trait among the smaller efforts.
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I initially tried to avoid diagonal adjacency too, but that proved too constraining on the surrounding fill.
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"Where we are working, [IoT] is a nice natural adjacency we can leverage," says MobileIron CEO Barry Mainz.
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It's worth remembering that their pre-KUWTK fame comes from their adjacency to the trial of the century.
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Speaking of adjacency ... People are using Facebook Live for all kinds of horrible things, including suicides and beatings.
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Ultimately, the user-submitted posts on Neighbors gain legitimacy from adjacency to verified crime reports submitted by Ring employees.
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For advertisers, the value proposition was a combination of content adjacency with a premium, influential audience consuming relevant content.
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" Altice is also mentioned as a potential merger candidate, in what the documents describe as an "opportunistic adjacency play.
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But that model of adjacency is irrelevant and increasingly superfluous in a world in which there is so much content.
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"We're hopeful the new Women's apparel layout with improved adjacency to Sephora will help stabilize and improve department comps," Chen said.
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There seems to be no logic governing the placement of one color to another, but the adjacency does not feel arbitrary either.
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In XCOM: Enemy Unknown and this year's XCOM 2, the strategy layer emphasizes adjacency bonuses and requires attention to longer term planning.
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Her father had his reservations about the job, with its late hours and adjacency to drinking, but she had her own goals.
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In that regard, "Rebels" had a decided advantage, strictly in terms of its adjacency to the first trilogy versus the latest one.
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However, the planet's adjacency to the star—it whizzes around once every 33 days—could produce intense space weather from the stellar activity.
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But OMD's Winkler argues the ad adjacency problem is more pronounced for YouTube than Facebook because ads run in between back-to-back videos.
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Hong Kong has become a testing ground for the Chinese e-wallet titans going global due to the city's geographic adjacency and cosmopolitan population.
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When I see people talking about how Linkin Park's music — specifically their lyrics — did a lot for them, it was by way of adjacency.
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"We argue increased movie ticket sales are an incremental traffic contributor for Cheesecake Factory, given the adjacency of its restaurants to movie theaters," he said.
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The other big adjacency that Overtime is growing into is sewing up the Gen Z generation for sports and growing with them as they age.
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I showed it to architects and they thought they could use it to show students the importance of adjacency to rooms is in designing a house.
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One of AutoLotto's investors, Adjacency Partners' Tom Arrix, formerly vice president of global marketing solutions at Facebook, said he sees international potential in the company, too.
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Julia Fine's novel is a wonderful addition to that genre of lyrical, poetic fantasies, akin to fairy tales in their delicacy and adjacency to the real world.
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YouTube said at the event that Johnson & Johnson — which initially pulled ads after the adjacency problems — would sponsor an upcoming talent competition series produced by Ryan Seacrest.
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And there's the idea that actual death is less proximate than living death, the drudge we live in where we've lost all sense of adjacency to others.
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The American entrepreneur has clients from Europe, North America and Asia, but he settled in the city for its adjacency to Southeast Asia and mainland China's massive market.
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He was disturbed by the lack of kindness toward his client, who had been traumatized by her adjacency to the rape, having herself been raped as a teenager.
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But the other important reason is "ad adjacency," which is ad industry speak for making sure your ads don't appear next to something offensive, like an ISIS video.
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But they withdrew from Wayfair even before they started selling on 1stdibs simply because they didn't want the adjacency of their brand with the lower-priced products on Wayfair.
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We started in the software team, building project management and collaboration tools for software developers, and over the last decade we've expanded into IT teams, sort of in adjacency.
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He identifies five digital dynamics (speed, adaptability, adjacency, scale, and precision) that have transformed the work of marketing, and proposes a relatively simple way of bringing order to the chaos.
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Content adjacency — or the images and videos that ads appear next to — has become a big issue for advertisers after reports showed ads running next to inappropriate content on YouTube.
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So she devotes the final third of her book to celebrating the uses of soft power, the vague, amorphous power of adjacency that accrues to the office of the first lady.
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Amazon has also met with several technology companies regarding how to prevent inappropriate content from appearing next to ads, an issue called content adjacency, several people with knowledge of the meetings told CNBC.
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But innocuous exchanges between DNC employees and their spouses or partners do not become evidence of corruption simply by virtue of their adjacency in a database to more substantive conversations about kneecapping Bernie Sanders.
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I wonder this too, and how — or if — "off-white adjacency" can be talked about in a way that acknowledges the experiences of those in my generation without seeming to mythologize or aggrandize them.
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"It was built here for a reason," with an L connection and adjacency to City Hall, a free interior public space as well as the plaza with its eye-catching Jean Dubuffet sculpture out front.
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You weigh the replies times X, the karma score of the user by Y, the adjacency of the person who tweeted by Z. You multiply that all together and you get the relevancy score of a particular comment.
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"It sits in our target market, which we have always been clear is below 150 seats, and will have natural adjacency to the E2 offering," he said, referring to the company's family of 80-120-seat regional jets.
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Finally, a significant nexus test is to be applied in cases where the water in question is not otherwise roped-in under federal regulation on account of its categorical inclusion as a tributary or due to the adjacency rule.
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Mr. Obama, born in 1961, is at the leading edge of the "generation" I was looking to define, and the journey that led to his becoming the first black president was impossible to separate from his adjacency to whiteness.
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That makes the energy storage products a nice business adjacency for Tesla — the company certainly has plenty of engineering expertise — but it also suggests that it's unlikely to be the main driver for Tesla in the next several years, or ever.
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A discomforting paradox lay beneath the whole confrontation, one that cut straight across the accepted modern vision of Asians and their adjacency to whiteness: If Liang (and, by extension, all Asian-Americans) enjoyed the protections of whiteness, then how do you explain his conviction?
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And in "As a Human Being," about a domestic altercation, the poem's obsession with pairs — repeated words and sentence structures — draws attention to the adjacency of a thing to its relative: the speaker to his father, his father to his mother, his mother to him.
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MMA has a complicated, bipolar history: the sport's adjacency to death was an early selling point ("two men enter, one man leaves") before it became a liability to run away from, and performance-enhancing drugs were a thinly veiled custom before they became a plague to wipe out.
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Though he is quick to acknowledge that members of our generation enjoy access to elements of white privilege — what he calls "off-white adjacency" — he explains it's important to take other factors into consideration when considering the successes of the Loving generation, namely economic class and the outsider-overachiever dynamic.
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YouTube CEO Susan Wojcicki said on Thursday during Brandcast, Google's annual presentation of upcoming content for advertisers, it was working on fixing the ad adjacency issue with more technology, giving brands more control over where their ads run and bringing in third-party companies like comScore and Integral Ad Science to monitor content.
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In a blog post on Monday, Google's chief business officer Philipp Schindler apologized for the ad adjacency issues, and said the company is working to make sure that ads only show up next to creators the company can vouch for, and make it easier for advertisers to control which kinds of content their ads appear next to.
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Although You Know You Want This may be timely in its occasional adjacency to #MeToo, its real canniness comes from apprehending the psychology not only of power, but of power-hunger as, itself, a form of weakness: how people harbor an impulse toward sadistic narcissism, and how little it takes for them to succumb to it.
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"We are dealing with a day and age where massive brand incursions can come in from any direction, be it a security breach, or be it negative adjacency (of advertising next to inappropriate content) but we are seeing CMOs really taking a step and saying: advertising is no longer advertising for advertising's sake," she told CNBC by phone.
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Google came under fire late last week when a report in The Times of London revealed ads from companies and charities including Mercedes-Benz, Waitrose and Marie Curie were running alongside neo-Nazi and jihadist videos on YouTube and websites served by the Google Display Network in the U.K. However, some agencies have suggested the ad adjacency issue is a digital advertising issue in general, especially in online video.
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The adjacency matrix may be used as a data structure for the representation of graphs in computer programs for manipulating graphs. The main alternative data structure, also in use for this application, is the adjacency list., p. 361: "There are two data structures that people often use to represent graphs, the adjacency list and the adjacency matrix.".
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A symmetric sparse matrix arises as the adjacency matrix of an undirected graph; it can be stored efficiently as an adjacency list.
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The main alternative to the adjacency list is the adjacency matrix, a matrix whose rows and columns are indexed by vertices and whose cells contain a Boolean value that indicates whether an edge is present between the vertices corresponding to the row and column of the cell. For a sparse graph (one in which most pairs of vertices are not connected by edges) an adjacency list is significantly more space-efficient than an adjacency matrix (stored as a two-dimensional array): the space usage of the adjacency list is proportional to the number of edges and vertices in the graph, while for an adjacency matrix stored in this way the space is proportional to the square of the number of vertices. However, it is possible to store adjacency matrices more space-efficiently, matching the linear space usage of an adjacency list, by using a hash table indexed by pairs of vertices rather than an array. The other significant difference between adjacency lists and adjacency matrices is in the efficiency of the operations they perform.
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An -adjacency matrix of a simple graph has if is an edge, if it is not, and on the diagonal. The Seidel adjacency matrix is a -adjacency matrix. This matrix is used in studying strongly regular graphs and two- graphs. The distance matrix has in position the distance between vertices and .
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The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros. The adjacency matrix of an empty graph is a zero matrix.
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In the CSR, all adjacencies of a vertex is sorted and compactly stored in a contiguous chunk of memory, with adjacency of vertex i+1 next to the adjacency of i. In the example on the left, there are two arrays, C and R. Array C stores the adjacency lists of all nodes. Array R stored the index in C, the entry R[i] points to the beginning index of adjacency lists of vertex i in array C. The CSR is extremely fast because it costs only constant time to access vertex adjacency. But it is only space- efficient for 1D partitioning.
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In an adjacency list, the neighbors of each vertex may be listed efficiently, in time proportional to the degree of the vertex. In an adjacency matrix, this operation takes time proportional to the number of vertices in the graph, which may be significantly higher than the degree. On the other hand, the adjacency matrix allows testing whether two vertices are adjacent to each other in constant time; the adjacency list is slower to support this operation.
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Using a naïve array implementation on a 32-bit computer, an adjacency list for an undirected graph requires about bytes of space, where is the number of edges of the graph. Noting that an undirected simple graph can have at most edges, allowing loops, we can let denote the density of the graph. Then, when , that is the adjacency list representation occupies more space than the adjacency matrix representation when . Thus a graph must be sparse enough to justify an adjacency list representation.
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Besides the space trade-off, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list. With an adjacency matrix, an entire row must instead be scanned, which takes time. Whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list.
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For use as a data structure, the main alternative to the adjacency list is the adjacency matrix. Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only bytes of contiguous space, where is the number of vertices of the graph. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a sparse graph, adjacency lists require less space, because they do not waste any space to represent edges that are not present.
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However, such advantage comes at a cost. index-free adjacency sacrifices the efficiency of queries that do not use graph traversals. Native graph databases use index-free adjacency to process CRUD operations on the stored data.
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In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A.Algebraic graph theory, by Norman L. Biggs, 1993, , p. 9 Some other similar mathematical objects are also called "adjacency algebra".
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An ancohemitonic scale is less dissonant than a cohemitonic scale, the count of their semitones being equal. In general, the number of semitones is more important to the perception of dissonance than the adjacency (or lack thereof) of any pair of them. Additional adjacency between semitones (once adjacency is present) does not necessarily increase the dissonance, the count of semitones again being equal.Keith, Michael. 1991.
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On the other hand, testing whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list..
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In comparative genomics, a sequence graph, also called an alignment graph, breakpoint graph, or adjacency graph, is a bidirected graph in which the vertices represent segments of DNA and the edges represent adjacency between segments in a genome. The segments are labeled by the DNA string they represent, and each edge connects the tail end of one segment with the head end of another segment. Each adjacency edge is labelled by a (possibly empty) string of DNA. Traversing a connected component of segments and adjacency edges (called a thread) yields a sequence, which typically represents a genome or a section of a genome.
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Properties of the adjacency algebra of G are associated with various spectral, adjacency and connectivity properties of G. Statement. The number of walks of length d between vertices i and j is equal to the (i, j)-th element of Ad. Statement. The dimension of the adjacency algebra of a connected graph of diameter d is at least d + 1. Corollary. A connected graph of diameter d has at least d + 1 distinct eigenvalues.
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An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering.
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The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582.
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6, pp. 126–127. A characterization of chordal bipartite graphs in terms of intersection graphs related to hypergraphs is given in. A bipartite graph is chordal bipartite if and only if its adjacency matrix is totally balanced if and only if the adjacency matrix is Gamma-free.
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Note that the adjacency matrix can be used to determine whether or not the graph is connected.
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A forest of stars is a set of star worlds whose adjacency matrix is a tree. This means that no intersecting star worlds create a cycle, or hole, in the overall space. If an object or space can be represented by a forest of stars, it can be mapped onto a sphere-world by mapping each star world onto the boundary of its parent star world in the adjacency tree. The root of an adjacency tree can be picked arbitrarily.
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Hope International is abetted by its adjacency to California State University, Fullerton, with which it has contract programs.
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In direct adjacency to the 5' UTR lies the cHP structure, which is essential for the viral replication.
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In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
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Three types of records are used: vertex records, edge records, and face records. Given a reference to an edge record, one can answer several types of adjacency queries (queries about neighboring edges, vertices and faces) in constant time. This kind of adjacency information is useful for algorithms such as Subdivision surface.
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It is also possible to store edge weights directly in the elements of an adjacency matrix. Besides the space tradeoff, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list, and takes time proportional to the number of neighbors. With an adjacency matrix, an entire row must instead be scanned, which takes a larger amount of time, proportional to the number of vertices in the whole graph.
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We can use this fact to calculate d_{DCJ}. Let N be the number of synteny blocks in genomes G_1 and G_2, C be the number of cycles in their adjacency graph, and I be the number of paths in their adjacency graph. Then d_{DCJ}(G_1, G_2)=N-(C+1/2).
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In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non- adjacent vertices. It is also called the Seidel matrix or--its original name-- the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by J. H. van Lint and J. J. Seidel in 1966 and extensively exploited by Seidel and coauthors.
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The SPB Link Metric sub-TLV occurs within the Multi-Topology Intermediate System Neighbor TLV or within the Extended IS Reachability TLV. Where this sub-TLV is not present for an IS-IS adjacency, then that adjacency will not carry SPB traffic for the given topology instance. There are multiple ECT algorithms defined for SPB; however, for the future, additional algorithms may be defined; similarly the SPB Adjacency Opaque Equal Cost Tree Algorithm TLV also occurs within the Multi-Topology Intermediate System TLV or the Extended IS Reachability TLV.
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Because BFS algorithm always uses the adjacency matrix as the representation of the graph. The natural 2D decomposition of matrix can also be an option to consider. In 2D partitioning, each processor has a 2D index (i,j). The edges and vertexes are assigned to all processors with 2D block decomposition, in which the sub- adjacency matrix is stored.
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Many graph problems are solved in the setting where the adjacency matrix or the adjacency list of the graph is streamed in some unknown order. There are also some problems that are very dependent on the order of the stream (i.e., asymmetric functions), such as counting the number of inversions in a stream and finding the longest increasing subsequence.
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IS-IS adjacency can be either broadcast or point-to-point. ;Hello Packet :The IS-IS hello packets needs to be exchanged periodically between 2 routers to establish adjacency. Based on the negotiation, one of them will be selected as DIS (Designated IS). This hello packet will be sent separately for Level-1 or Level-2.
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Trivial Graph Format (TGF) is a simple text-based adjacency list file format for describing graphs, widely used because of its simplicity.
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For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix A of the graph, if Q is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries are unity except the diagonal entries which are zero), then the adjacency matrix of the complement of A is Q-A. The complement is not defined for multigraphs.
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Instead of using random accesses to construct L(t) the file H is used. # Perform a parallel scan of sorted list L(t-1) and H. Extract the adjacency lists for nodes v\in L(t-1), that can be found in H. # The adjacency lists for the remaining nodes that could not be found in H need to be fetched. A scan over L(t-1) yields the partition identifiers. After sorting and deletion of duplicates, the respective files F_i can be concatenated into a temporary file F'. # The missing adjacency lists can be extracted from F' with a scan.
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The proof is bijective: a matrix is an adjacency matrix of a DAG if and only if is a (0,1) matrix with all eigenvalues positive, where denotes the identity matrix. Because a DAG cannot have self-loops, its adjacency matrix must have a zero diagonal, so adding preserves the property that all matrix coefficients are 0 or 1., Article 04.3.3.
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The Seidel matrix of G is also the adjacency matrix of a signed complete graph KG in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and KG. The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graphs.
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Let I denote the identity matrix and let J denote the matrix of ones, both matrices of order v. The adjacency matrix A of a strongly regular graph satisfies two equations. First: :AJ = JA = kJ, which is a trivial restatement of the regularity requirement. This shows that k is an eigenvalue of the adjacency matrix with the all-ones eigenvector.
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SNA software generates these features from raw network data formatted in an edgelist, adjacency list, or adjacency matrix (also called sociomatrix), often combined with (individual/node-level) attribute data. Though the majority of network analysis software uses a plain text ASCII data format, some software packages contain the capability to utilize relational databases to import and/or store network features.
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In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e.
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Moreover, every permutation of the index set leads to a natural isomorphism; they are uncountably many! Another example. A structure of a (simple) graph on a set V = of vertices may be described by means of its adjacency matrix, a (0,1)-matrix of size n×n (with zeros on the diagonal). More generally, for arbitrary V an adjacency function on V × V may be used.
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In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number.
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Sausalito has a Mediterranean climate (Köppen climate classification Csb) with far lower temperatures than expected because of its adjacency to San Francisco Bay and the resultant onshore breezes.
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Not all graph families have local structures. For some families, a simple counting argument proves that adjacency labeling schemes do not exist: only bits may be used to represent an entire graph, so a representation of this type can only exist when the number of -vertex graphs in the given family is at most . Graph families that have larger numbers of graphs than this, such as the bipartite graphs or the triangle-free graphs, do not have adjacency labeling schemes.. However, even families of graphs in which the number of graphs in the family is small might not have an adjacency labeling scheme; for instance, the family of graphs with fewer edges than vertices has -vertex graphs but does not have an adjacency labeling scheme, because one could transform any given graph into a larger graph in this family by adding a new isolated vertex for each edge, without changing its labelability. Kannan et al.
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If there are in total P=R·C processors, then the adjacency matrix will be divided like below: 2D-partition of the adjacency matrix. There are C columns and R·C block rows after this division. For each processor, they are in charge of C blocks, namely the processor (i,j) stores Ai,j(1) to Ai,j(C) blocks. The conventional 1D partitioning is equivalent to the 2D partitioning with R=1 or C=1.
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For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in binary. Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.
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For d-regular graphs which are pseudo-random in the sense that their adjacency matrix has second largest eigenvalue (in absolute value) at most , d is the optimal number of colors .
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On 24 February 2000, personnel from the two nations encountered each other in Toledo District. The two countries held further talks on 14 March 2000, at the Organization of American States (OAS) in the presence of Secretary General César Gaviria at OAS headquarters in Washington, D.C. Eventually they agreed to establish an "adjacency zone" extending one kilometre (0.62 mi) on either side of the 1859 treaty line, now designated the "adjacency line", and to continue negotiations.
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The theory was introduced, carefully elaborated and applied by the Canadian group of people around André Joyal. The power of the theory comes from its level of abstraction. The "description format" of a structure (such as adjacency list versus adjacency matrix for graphs) is irrelevant, because species are purely algebraic. Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.
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The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties.
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Similarly, taste → touch synesthesia may arise from connections between gustatory regions and regions of the somatosensory system involved in processing touch. However, not all forms of synesthesia are easily explained by adjacency.
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The adjacency matrix of a directed graph is unique up to identical permutation of rows and columns. Another matrix representation for a directed graph is its incidence matrix. See direction for more definitions.
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The graph application is for manipulating directed and undirected graphs. Some the standard graph functions exist (like for adjacency and cliques) together with combinatorial functions like computing the lattice represented by a directed acyclic graph.
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The first-order language of graphs is the collection of well-formed sentences in mathematical logic formed from variables representing the vertices of graphs, universal and existential quantifiers, logical connectives, and predicates for equality and adjacency of vertices. For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence :\forall u:\exists v: u\sim v where the \sim symbol indicates the adjacency relation between two vertices., Section 1.2, "What Is a First Order Theory?", pp. 15–17.
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The main operation performed by the adjacency list data structure is to report a list of the neighbors of a given vertex. Using any of the implementations detailed above, this can be performed in constant time per neighbor. In other words, the total time to report all of the neighbors of a vertex v is proportional to the degree of v. It is also possible, but not as efficient, to use adjacency lists to test whether an edge exists or does not exist between two specified vertices.
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In the analysis of algorithms, the input to breadth-first search is assumed to be a finite graph, represented explicitly as an adjacency list, adjacency matrix, or similar representation. However, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. In this context, a search method is described as being complete if it is guaranteed to find a goal state if one exists. Breadth-first search is complete, but depth-first search is not.
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802.1aq has been designed to operate in parallel with other network layer protocols such as IPv4 and IPv6; therefore, failure of two nodes to establish an SPB adjacency will not cause network layer protocols to also reject an adjacency. 802.1aq has been assigned the Network Layer Protocol ID (NLPID) value 0xC1, as per RFC 6328, and is used by SPB Bridges to indicate their ability to form adjacencies and operate as part of a 802.1aq domain. 802.1aq frames flow on adjacencies that advertise this NLPID in both directions, and nodes regard an adjacency that has not advertised in both directions as non- existent (infinite link metric). 802.1aq augments the normal IIH PDU with three new TLVs, which like all other SPB TLVs, travel within Multi-Topology TLVs, therefore allowing multiple logical instances of SPB within a single IS- IS protocol instance.
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The blue lines under the castle was not part of the old coat of arms. They illustrate the municipality's and the castle's adjacency to water. The gate is open, to illustrate hospitality.Nyborg.dk "Designguide" Retrieved 31 July 2020Danskekommuner.
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The Aanderaa–Karp–Rosenberg conjecture concerns implicit graphs given as a set of labeled vertices with a black-box rule for determining whether any two vertices are adjacent. This definition differs from an adjacency labeling scheme in that the rule may be specific to a particular graph rather than being a generic rule that applies to all graphs in a family. Because of this difference, every graph has an implicit representation. For instance, the rule could be to look up the pair of vertices in a separate adjacency matrix.
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Combined with this is single point provisioning for logical connectivity membership. IS-IS is therefore augmented with a small number of TLVs and sub-TLVs, and supports two Ethernet encapsulating data paths, 802.1ad Provider Bridges (PB) and 802.1ah Provider Backbone Bridges (PBB). SPB is designed to run in parallel with other network layer protocols such as IPv4 and IPv6. Standards mandate that the failure of two nodes to establish an SPB adjacency will not have collateral impact, such as the rejection of an adjacency for other network layer protocols (e.g. OSPF).
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Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs have equal multisets of eigenvalues. enneahedra, the smallest possible cospectral polyhedral graphs Cospectral graphs need not be isomorphic, but isomorphic graphs are always cospectral.
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The segments can be thought of as synteny blocks, with the edges dictating how to arrange these blocks in a particular genome, and the labelling of the adjacency edges representing bases that are not contained in synteny blocks.
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Next, the remaining adjacency lists are merged into H with a single pass. # A(t) is created by a simple scan. The partition information is attached to each node in A(t). # The algorithm proceeds like the MR algorithm.
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In graph theory, a fractional isomorphism of graphs whose adjacency matrices are denoted A and B is a doubly stochastic matrix D such that DA = BD. If the doubly stochastic matrix is a permutation matrix, then it constitutes a graph isomorphism.
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In the case of a shared memory model, the graph representations used for parallel processing are the same as in the sequential case, since parallel read-only access to the graph representation (e.g. an adjacency list) is efficient in shared memory.
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In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The degree matrix indicates the degree of vertices. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph. The distance matrix, like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a shortest path between two vertices.
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Provided the graph is described using an adjacency list, Kosaraju's algorithm performs two complete traversals of the graph and so runs in Θ(V+E) (linear) time, which is asymptotically optimal because there is a matching lower bound (any algorithm must examine all vertices and edges). It is the conceptually simplest efficient algorithm, but is not as efficient in practice as Tarjan's strongly connected components algorithm and the path-based strong component algorithm, which perform only one traversal of the graph. If the graph is represented as an adjacency matrix, the algorithm requires Ο(V2) time.
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Data lookup performance is dependent on the access speed from one particular node to another. Because index-free adjacency enforces the nodes to have direct physical RAM addresses and physically point to other adjacent nodes, it results in a fast retrieval. A native graph system with index-free adjacency does not have to move through any other type of data structures to find links between the nodes. Directly related nodes in a graph are stored in the cache once one of the nodes are retrieved, making the data lookup even faster than the first time a user fetches a node.
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In the context of this article, all graphs will be simple and undirected, unless stated otherwise. This means that the edges of the graph form a set (and not a multiset) and each edge is a pair of distinct vertices. Graphs are assumed to have an implicit representation in which each vertex has a unique identifier or label and in which it is possible to test the adjacency of any two vertices, but for which adjacency testing is the only allowed primitive operation. Informally, a graph property is a property of a graph that is independent of labeling.
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A graph may be fully specified by its adjacency matrix A, which is an nxn square matrix, with Aij specifying the nature of the connection between vertex i and vertex j. For a simple graph, Aij= 0 or 1, indicating disconnection or connection respectively, with Aii=0. Graphs with self-loops will be characterized by some or all Aii being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all Aij being equal to a positive integer. Undirected graphs will have a symmetric adjacency matrix (Aij=Aji).
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Admiralty Point was a government naval reserve, and was thus saved from development. The area is now a regional park. Belcarra's adjacency to several islands makes it a desired spot for boaters. Bedwell Bay and Sasamat Lake also are nearby, increasing the appeal.
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Knowledge about the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory. The genus of a group is the minimum genus for any Cayley graph of that group.
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In 2003, this institute separated from Shahid Chamran University and became Khorramshahr University of Marine Science and Technology (KMSU), with access to open waters, the Shatt al-Arab and Bahmanshir rivers, and adjacency to the Shadegan and Hor Swamps, situated in the Arvand Free Zone.
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The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network. This allows for the analysis of time series by network measures. Applications range from detection of regime changes over characterizing dynamics to synchronization analysis.
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This gives the player reserves in times of shortage or allows them to stockpile resources. If the resource generation exceeds the player's capacity, the material is wasted. An adjacency system allows certain structures to benefit from being built directly adjacent to others.Rorie (2007). p.
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When the graph is stored in the form of adjacency list or matrix, priority queue can be used to extract minimum efficiently when implementing Dijkstra's algorithm, although one also needs the ability to alter the priority of a particular vertex in the priority queue efficiently.
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For instance, the trpA and trpB genes in Escherichia coli encode the two subunits of the tryptophan synthase enzyme known to interact to catalyze a single reaction. The adjacency of these two genes was shown to be conserved across nine different bacterial and archaeal genomes.
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In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.
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A lollipop graph may also be understood, especially in spectral graph theory, as the coalescence of a cycle and a path with pendant vertex as distinguished vertex. In this case, it is known that this class of graphs is determined by its adjacency and laplacian spectra.
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By explicitly constructing the complete graph on n vertices, which has n(n-1)/2 edges, a rectilinear minimum spanning tree can be found using existing algorithms for finding a minimum spanning tree. In particular, using Prim's algorithm with an adjacency matrix yields time complexity O(n2).
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The deformed Laplacian is commonly defined as :\Delta(s) = I - sA + s^2(D - I) where I is the unit matrix, A is the adjacency matrix, and D is the degree matrix, and s is a (complex-valued) number. The standard Laplacian is just \Delta(1).
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For instance, a Car class can compose a Wheel one. In the object graph a Car instance will have up to four links to its wheels, which can be named frontLeft, frontRight, backLeft and backRight. Example of an adjacency list representation: c:Car → {frontLeft:Wheel, frontRight:Wheel, backLeft:Wheel, backRight:Wheel}.
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In the mathematical field of graph theory, the degree matrix is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex.. It is used together with the adjacency matrix to construct the Laplacian matrix of a graph..
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The station was moved underground on March 2, 2019 to reduce traffic congestion from level crossings on Sanayo-doro Avenue. On March 14, 2020, the station was renamed to . The name was changed because of the station's adjacency to the Daishi Bridge linking Kawasaki City to Tokyo's Ōta Ward.
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University Park comprises multiple neighborhoods and associations and contains a high concentration of community facilities and amenities. In addition to its greenbelts and bikeways, the village contains community facilities, parks, schools, religious institutions, a library, commercial centers, adjacency to open space and recreational areas, and access to transportation.
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To prove that transitive reduction is as easy as transitive closure, Aho et al. rely on the already-known equivalence with Boolean matrix multiplication. They let A be the adjacency matrix of the given directed acyclic graph, and B be the adjacency matrix of its transitive closure (computed using any standard transitive closure algorithm). Then an edge uv belongs to the transitive reduction if and only if there is a nonzero entry in row u and column v of matrix A, and there is a zero entry in the same position of the matrix product AB. In this construction, the nonzero elements of the matrix AB represent pairs of vertices connected by paths of length two or more.
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Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation. List structures include the edge list, an array of pairs of vertices, and the adjacency list, which separately lists the neighbors of each vertex: Much like the edge list, each vertex has a list of which vertices it is adjacent to. Matrix structures include the incidence matrix, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix, in which both the rows and columns are indexed by vertices.
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In graph theory, a graph or digraph whose adjacency matrix is circulant is called a circulant graph (or digraph). Equivalently, a graph is circulant if its automorphism group contains a full-length cycle. The Möbius ladders are examples of circulant graphs, as are the Paley graphs for fields of prime order.
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Global approaches rely on properties of the entire graph and do not rely on an arbitrary initial partition. The most common example is spectral partitioning, where a partition is derived from approximate eigenvectors of the adjacency matrix, or spectral clustering that groups graph vertices using the eigendecomposition of the graph Laplacian matrix.
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A GIS can recognize and analyze the spatial relationships that exist within digitally stored spatial data. These topological relationships allow complex spatial modelling and analysis to be performed. Topological relationships between geometric entities traditionally include adjacency (what adjoins what), containment (what encloses what), and proximity (how close something is to something else).
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The bulk of the game is played out on duplicate boards, one per player, with full knowledge of friendly forces but limited knowledge of the opponent's. Knowledge of opposing forces is gained by air and sea reconnaissance. Unlike the 1964 version, the search boards use hexagonal spaces to remove questions of corner adjacency.
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View of Mulyanka from the bridge at Krasin Street Mulyanka runs along the borders of the city of Perm and is contaminated by industrial waste, such at that of stock-breeding farms and sawmills. Mulyanka is influenced by the adjacency of oil refinery of Lukoil-Permnefteorgsintez which was put into operation in 1958.
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Relevant next turn refers to the initiation or invitation to respond between speakers, including verbal and nonverbal prompts for turn-taking in conversation. Questions and answers act as adjacency pairs,Sacks, H., Schegloff, E. A., & Jefferson, G. (1974). "A simplest systematics for the organization of turn-taking for conversation." Language, 50, 696-735.
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Chalillo Hydroelectric Dam is situated in the westernmost part of the country where the Macal River converges with the Raspaculo River. It is located in the Maya Mountains and shares adjacency with the Chiquibul National Park and the Pine Ridge Forest Reserve. Positioning Chalillo Hydroelectric Dam approximately 16 kilometers from the Guatemalan border.
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These facts were observed by . If G is a forest, then its matching polynomial is equal to the characteristic polynomial of its adjacency matrix. If G is a path or a cycle, then MG(x) is a Chebyshev polynomial. In this case μG(1,x) is a Fibonacci polynomial or Lucas polynomial respectively.
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Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only 2/8 bytes to represent a directed graph, or (by using a packed triangular format and only storing the lower triangular part of the matrix) approximately 2/16 bytes to represent an undirected graph. Although slightly more succinct representations are possible, this method gets close to the information- theoretic lower bound for the minimum number of bits needed to represent all -vertex graphs.. For storing graphs in text files, fewer bits per byte can be used to ensure that all bytes are text characters, for instance by using a Base64 representation.. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a large sparse graph, adjacency lists require less storage space, because they do not waste any space to represent edges that are not present. An alternative form of adjacency matrix (which, however, requires a larger amount of space) replaces the numbers in each element of the matrix with pointers to edge objects (when edges are present) or null pointers (when there is no edge).
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The kth power of a graph with n vertices and m edges may be computed in time O(mn) by performing a breadth first search starting from each vertex to determine the distances to all other vertices, or slightly faster using more sophisticated algorithms. Alternatively, If A is an adjacency matrix for the graph, modified to have nonzero entries on its main diagonal, then the nonzero entries of Ak give the adjacency matrix of the kth power of the graph,. from which it follows that constructing kth powers may be performed in an amount of time that is within a logarithmic factor of the time for matrix multiplication. The kth powers of trees can be recognized in time linear in the size of the input graph. .
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Kursky is connected to the Lengradskiy Line from the other side, enabling long-distance trains from St. Petersburg going on to other cities to pass through Russia's capital. Because of its three directions, its adjacency to the city center, and its connection to three major metro lines, Kursky is one of Moscow's busiest railway stations.
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The third term gives the corresponding value when i and j are not connected. Since the three cases are mutually exclusive and collectively exhaustive, the simple additive equality follows. Conversely, a graph whose adjacency matrix satisfies both of the above conditions and which is not a complete or null graph is a strongly regular graph.
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48(3):292-306, 2001. Static DSMs represent systems where all of the elements exist simultaneously, such as components of a machine or groups in an organization. A static DSM is equivalent to an N2 chart or an adjacency matrix. The marking in the off- diagonal cells is often largely symmetrical to the diagonal (e.g.
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Geometry shader programs are executed after vertex shaders. They take as input a whole primitive, possibly with adjacency information. For example, when operating on triangles, the three vertices are the geometry shader's input. The shader can then emit zero or more primitives, which are rasterized and their fragments ultimately passed to a pixel shader.
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The permanent of a (0,1)-matrix is equal to the number of vertex-disjoint cycle covers of a directed graph with this adjacency matrix. This fact is used in a simplified proof showing that computing the permanent is #P-complete.Ben-Dor, Amir and Halevi, Shai. (1993). "Zero-one permanent is #P-complete, a simpler proof".
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Thus, we want to describe all the cells of the subdivision, plus all the incidence and adjacency relations between these cells. When all the represented cells are simplexes, a simplicial complex may be used, but when we want to represent any type of cells, we need to use cellular topological models like combinatorial maps or generalized maps.
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The degree sequence can be drawn from a theoretical distribution or it can represent a real network (determined from the adjacency matrix of the network). # Choose two stubs uniformly at random and connect them to form an edge. Choose another pair from the remaining 2m-2 stubs and connect them. Continue until you run out of stubs.
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The Hercules Family is a group of constellations connected mainly by their adjacency on the celestial sphere. It is Menzel's largest grouping, and extends from declination +60° to −70°, mostly in the western hemisphere. It includes Hercules, Sagitta, Aquila, Lyra, Cygnus, Vulpecula, Hydra, Sextans, Crater, Corvus, Ophiuchus, Serpens, Scutum, Centaurus, Lupus, Corona Australis, Ara, Triangulum Australe, and Crux.
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For Sale: Condo with Chicken Coop, Sara Schaeffer Munoz, in The Wall Street Journal, May 17, 2007. The value of each parcel is increased by adjacency and access to the conserved land, which allows development density significantly below that allowed by zoning.Organic Farms as Subdivision Amenities, Alec Appelbaum in The New York Times, July 1, 2009.
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Entries congruent modulo N to 0, 1, or N − 1 do not appear in this sequence of numbers,. See Section 2. because they would correspond either to a loop or multiple adjacency, neither of which are permitted in simple graphs. Often the pattern repeats, and the number of repetitions can be indicated by a superscript in the notation.
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For many natural P-complete graph problems, where the graph is expressed in a natural representation such as an adjacency matrix, solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller; but this requires nontrivial proof, since succinct circuits can only describe a subclass of graphs.Papadimitriou (1994), section 20.1, page 492.
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The problem can also be stated in terms of symmetric matrices of zeros and ones. The connection can be seen if one realizes that each graph has an adjacency matrix where the column sums and row sums correspond to (d_1,\ldots,d_n). The problem is then sometimes denoted by symmetric 0-1-matrices for given row sums.
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Suppose two directed or undirected graphs and with adjacency matrices and are given. and are isomorphic if and only if there exists a permutation matrix such that : P A_1 P^{-1} = A_2. In particular, and are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore serve as isomorphism invariants of graphs.
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It may be given an adjacency labeling scheme in which the points that are endpoints of line segments are numbered from 1 to 2n and each vertex of the graph is represented by the numbers of the two endpoints of its corresponding interval. With this representation, one may check whether two vertices are adjacent by comparing the numbers that represent them and verifying that these numbers define overlapping intervals. The same approach works for other geometric intersection graphs including the graphs of bounded boxicity and the circle graphs, and subfamilies of these families such as the distance-hereditary graphs and cographs. However, a geometric intersection graph representation does not always imply the existence of an adjacency labeling scheme, because it may require more than a logarithmic number of bits to specify each geometric object.
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More generally, all -vertex cliques can be listed by a similar algorithm that takes time proportional to the number of edges multiplied by the arboricity to the power . For graphs of constant arboricity, such as planar graphs (or in general graphs from any non-trivial minor-closed graph family), this algorithm takes time, which is optimal since it is linear in the size of the input.. If one desires only a single triangle, or an assurance that the graph is triangle- free, faster algorithms are possible. As observe, the graph contains a triangle if and only if its adjacency matrix and the square of the adjacency matrix contain nonzero entries in the same cell. Therefore, fast matrix multiplication techniques such as the Coppersmith–Winograd algorithm can be applied to find triangles in time .
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This is the lowest state of a router that may be considered as a Designated Router. #ExStart: The ExStart state is the first step of adjacency of two routers. #Exchange: In the Exchange state, a router is sending its link-state database information to the adjacent neighbor. At this state, a router is able to exchange all OSPF routing protocol packets.
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The just perfect fifth, together with the octave, forms the basis of Pythagorean tuning. A slightly narrowed perfect fifth is likewise the basis for meantone tuning. The circle of fifths is a model of pitch space for the chromatic scale (chromatic circle), which considers nearness as the number of perfect fifths required to get from one note to another, rather than chromatic adjacency.
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Another method for finding community structures in networks is hierarchical clustering. In this method one defines a similarity measure quantifying some (usually topological) type of similarity between node pairs. Commonly used measures include the cosine similarity, the Jaccard index, and the Hamming distance between rows of the adjacency matrix. Then one groups similar nodes into communities according to this measure.
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Wings 3D is a free and open-source subdivision modeler inspired by Nendo and Mirai from Izware. Wings 3D is named after the winged-edge data structure it uses internally to store coordinate and adjacency data, and is commonly referred to by its users simply as Wings. Wings 3D is available for Windows, Linux, and Mac OS X, using the Erlang environment.
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This was used as a horse market (due to its adjacency to the royal stables), but also as an official square for royal and religious ceremonies. It is occupied today by a large roundabout known as the Citadel Square (Midan al-Qal'a) or Saladin Square (Midan Salah ad-Din) next to which are the massive mosques of Sultan Hassan and al-Rifa'i.
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It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the category-theoretic product in the category of graphs and graph homomorphisms. However it is actually the Kronecker tensor product of the adjacency matrices of the graphs. Compare also the section Tensor product of linear maps above.
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Because the facility is adjacent to Folsom State Prison (FSP), both SAC and FSP share the mailing address: Represa, CA 95671. Represa (translated as "dam" from the Spanish language) is the name given to the State Prison post office because its adjacency to Folsom Dam.Durham, David L. California's Geographic Names. A Gazetteer of Historic and Modern Names of the State.
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An alternative is adjacency diagrams, which is a space- filling variant of the node-link diagram. Instead of drawing a link between hierarchies, nodes are drawn as solid areas with sub-sections inside of each section. This method allows for size to be easily represented than in the node-link diagrams. Enclosure diagrams are also a space-filling visualization method.
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There are also other approaches wherein the groups are defined basing on, e.g., physical adjacency of rows. For example, Infobright implemented a database engine wherein data was partitioned onto rough rows, each consisting of 64K of physically consecutive (or almost consecutive) rows. Rough rows were automatically labeled with compact information about their values on data columns, often involving multi-column and multi-table relationships.
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Note that t is always labelled with the value n+1 and s is always labelled with 0. The depth-first traversal is then repeated, but this time the adjacency list of each vertex is visited from right-to-left. When completed, s and t, and their incident edges, are removed. Each remaining vertex stores a 2-dimensional label with values from 1 to n.
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A red–black tree plotted by Graphviz. Undirected graph showing adjacency of the 48 contiguous United States Graphviz (short for Graph Visualization Software) is a package of open-source tools initiated by AT&T; Labs Research for drawing graphs specified in DOT language scripts. It also provides libraries for software applications to use the tools. Graphviz is free software licensed under the Eclipse Public License.
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GraphDB had index-free adjacency, which meant that it not necessary to manage a global index for relationships between nodes/entities. The linked objects contained direct reference to their adjacent neighboring nodes. The sones graph database was able to store and retrieve unstructured properties in any node of the graph. The idea was also to transfer unstructured data to structured data and vice versa.
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Cohen et al. found that of a pair of co-expressed genes only one promoter has an Upstream Activating Sequence (UAS) associated with that expression pattern. They suggested that UASs can activate genes that are not in immediate adjacency to them. This explanation could explain the co-expression of small clusters, but many clusters contain to many genes to be regulated by a single UAS.
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The Ratio and Algebra attribute hierarchy can also be expressed in matrix form. To begin, the direct relationship among the attributes is specified by a binary adjacency matrix (A) of order (k,k), where k is the number of attributes, such that each element in the A matrix represents the absence (i.e., 0) or presence (i.e., 1) of a direct connection between two attributes.
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This anti-agreement is standard in Modern Welsh - excepting the adjacency condition. It had general usage in Old Welsh and therefore, presumably, in Cumbric.. It has also been argued that this was a language-internal development that arose during the Middle English period. The lack of northern texts in Old English means that explaining the origin of the rule with any degree of certainty is difficult.
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The adjacency of water gives a number of distinctive characteristics to littoral regions. The erosive power of water results in particular types of landforms, such as sand dunes, and estuaries. The natural movement of the littoral along the coast is called the littoral drift. Biologically, the ready availability of water enables a greater variety of plant and animal life, and particularly the formation of extensive wetlands.
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Graphs and directed graphs can be viewed as a special case of the far more general notion called relational structures (defined as a set with a tuple of relations on it). Directed graphs are structures with a single binary relation (adjacency) on the domain (the vertex set)., note relational structures are called relational systems there. Under this view, homomorphisms of such structures are exactly graph homomorphisms.
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GraphBLAS () is an API specification that defines standard building blocks for graph algorithms in the language of linear algebra. GraphBLAS is built upon the notion that a sparse matrix can be used to represent graphs as either an adjacency matrix or an incidence matrix. The GraphBLAS specification describes how graph operations (e.g. traversing and transforming graphs) can be efficiently implemented via linear algebraic methods (e.g.
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Julio René Alvarado Ruano (2002?-20 April 2016) was a fourteen-year-old Guatemalan male who was killed in the Belize–Guatemala adjacency zone during an armed confrontation between the Belize Defence Force, Belizean park rangers, and Guatemalan farmers. The incident raised tensions between Belize and Guatemala, who have an ongoing border dispute since 1821. Belizean officials maintain that the Belize Defence Force acted in self-defence.
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BFD may be used on many different underlying transport mechanisms and layers, and operates independently of all of these. Therefore, it needs to be encapsulated by whatever transport it uses. For example, monitoring MPLS LSPs involves piggybacking session establishment on LSP-Ping packets. Protocols that support some form of adjacency setup, such as OSPF, IS-IS, BGP or RIP may also be used to bootstrap a BFD session.
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The neighbor relationship table is called an adjacency database. Two OSPF routers are neighbors if they are members of the same subnet and share the same area ID, subnet mask, timers and authentication. In essence, OSPF neighborship is a relationship between two routers that allow them to see and understand each other but nothing more. OSPF neighbors do not exchange any routing information – the only packets they exchange are Hello packets.
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In an adjacency list in which the neighbors of each vertex are unsorted, testing for the existence of an edge may be performed in time proportional to the minimum degree of the two given vertices, by using a sequential search through the neighbors of this vertex. If the neighbors are represented as a sorted array, binary search may be used instead, taking time proportional to the logarithm of the degree.
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The Biggs-Smith graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003 The characteristic polynomial of the Biggs-Smith graph is : (x-3) (x-2)^{18} x^{17} (x^2-x-4)^9 (x^3+3 x^2-3)^{16}.
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The shown graph appears as a subgraph of a graph G if, and only if. G satisfies the formula \exists x_1, x_2, x_3, x_4. x_1 \sim x_2 \land x_2 \sim x_3 \land x_3 \sim x_1 \land x_3 \sim x_4. In the first-order logic of graphs, a graph property is expressed as a quantified logical formula whose variables represent graph vertices, with predicates for equality and adjacency testing.
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For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence :\forall u\exists v(u\sim v) where the \sim symbol indicates the undirected adjacency relation between two vertices. This sentence can be interpreted as meaning that for every vertex u there is another vertex v that is adjacent to u., Section 1.2, "What Is a First Order Theory?", pp. 15–17.
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Let the adjacency matrix for the network be represented by A, where A_=0 means there's no edge (no interaction) between nodes v and w and A_{vw} = 1 means there is an edge between the two. Also for simplicity we consider an undirected network. Thus A_{vw} = A_{wv}. (It is important to note that multiple edges may exist between two nodes, but here we assess the simplest case).
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The problem can also be stated in terms of zero-one matrices. The connection can be seen if one realizes that each directed graph has an adjacency matrix where the column sums and row sums correspond to (a_1,\cdots,a_n) and (b_1,\ldots,b_n). Note that the diagonal of the matrix only contains zeros. The problem is then often denoted by 0-1-matrices for given row and column sums.
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The theorem has particular use in algebraic graph theory. The "underlying graph" of a nonnegative n-square matrix is the graph with vertices numbered 1, ..., n and arc ij if and only if Aij ≠ 0. If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the adjacency matrix of a strongly connected graph is irreducible.
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The distance is the length of a shortest path connecting the vertices. Unless lengths of edges are explicitly provided, the length of a path is the number of edges in it. The distance matrix resembles a high power of the adjacency matrix, but instead of telling only whether or not two vertices are connected (i.e., the connection matrix, which contains boolean values), it gives the exact distance between them.
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For instance, representing a graph as a unit disk graph may require exponentially many bits for the coordinates of the disk centers.. ;Low-dimensional comparability graphs: The comparability graph for a partially ordered set has a vertex for each set element and an edge between two set elements that are related by the partial order. The order dimension of a partial order is the minimum number of linear orders whose intersection is the given partial order. If a partial order has bounded order dimension, then an adjacency labeling scheme for the vertices in its comparability graph may be defined by labeling each vertex with its position in each of the defining linear orders, and determining that two vertices are adjacent if each corresponding pair of numbers in their labels has the same order relation as each other pair. In particular, this allows for an adjacency labeling scheme for the chordal comparability graphs, which come from partial orders of dimension at most four...
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However, an algorithm that is given as input an implicit graph of this type must operate on it only through the implicit adjacency test, without reference to how the test is implemented. A graph property is the question of whether a graph belongs to a given family of graphs; the answer must remain invariant under any relabeling of the vertices. In this context, the question to be determined is how many pairs of vertices must be tested for adjacency, in the worst case, before the property of interest can be determined to be true or false for a given implicit graph. Rivest and Vuillemin proved that any deterministic algorithm for any nontrivial graph property must test a quadratic number of pairs of vertices.. The full Aanderaa–Karp–Rosenberg conjecture is that any deterministic algorithm for a monotonic graph property (one that remains true if more edges are added to a graph with the property) must in some cases test every possible pair of vertices.
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In the mathematical field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. The 80-edge variant is the order-5 halved cube graph; it was called the Clebsch graph name by Seidel (1968)J. J. Seidel, Strongly regular graphs with (−1,1,0) adjacency matrix having eigenvalue 3, Lin. Alg. Appl. 1 (1968) 281-298.
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In 1974, Biggs published Algebraic Graph Theory which articulates properties of graphs in algebraic terms, then works out theorems regarding them. In the first section, he tackles the applications of linear algebra and matrix theory; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. Next, there is and wide-ranging description of the theory of chromatic polynomials. The last section discusses symmetry and regularity properties.
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The primary technique used to control a flood fill will either be data-centric or process-centric. A data-centric approach can use either a stack or a queue to keep track of seed pixels that need to be checked. A process-centric algorithm must necessarily use a stack. A 4-way flood-fill algorithm that uses the adjacency technique and a queue as its seed pixel store yields an expanding lozenge-shaped fill.
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NodeXL is intended for users with little or no programming experience to allow them to collect, analyze, and visualize a variety of networks. NodeXL integrates into Microsoft Excel 2007, 2010, 2013 and 2016 and opens as a workbook with a variety of worksheets containing the elements of a graph structure such as edges and nodes. NodeXL can also import a variety of graph formats such as edgelists, adjacency matrices, GraphML, UCINet .dl, and Pajek .net.
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The Rancho has traditionally been represented by the Burbank Rancho Homeowners, which was formed in 1963 by Floran Frank and other equestrian enthusiasts and is the oldest neighborhood group in the city. The community recently stopped the development of a Whole Foods store in the Rancho area. Rancho real estate sells at a premium due to its equestrian zoning, numerous parks, connection to riding trails in Griffith Park and its adjacency to Warner Bros.
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Thus, the connectivity can in general not be described by a weight or connectivity matrix but rather as an adjacency list. To manipulate or observe the network dynamics, the experimenter can define so- called devices which represent the various instruments (for measuring and stimulation) found in an experiment. These devices write their data either to memory or to file. NEST is extensible and new models for neurons, synapses, and devices can be added.
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Median graphs have a close connection to the solution sets of 2-satisfiability problems that can be used both to characterize these graphs and to relate them to adjacency-preserving maps of hypercubes., Proposition 2.5, p.8; ; ; , Theorem S, p. 72. A 2-satisfiability instance consists of a collection of Boolean variables and a collection of clauses, constraints on certain pairs of variables requiring those two variables to avoid certain combinations of values.
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This corrective process is limited, however, since increasing the population of activated neurons can result in off-target activation of neighboring circuitry. This is often culminated in undesired motor side effects upon stimulation, due to the adjacency of the MFB to the internal capsule, a bundle of axons carrying descending motor information to the brainstem. Inadvertent stimulation of these axons can lead to motor output such as movement of the head or paw twitching.
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Missing topics include the symmetries of graphs, cliques, connections between graphs and linear algebra including adjacency matrices, algebraic graph theory and spectral graph theory, the connectivity of graphs (or even its Biconnected components), Hall's marriage theorem, line graphs, interval graphs, and the theory of tournaments. There is also only a single chapter of coverage on algorithms and real-world applications of graph theory. Additionally, the book skips over "difficult or long proofs".
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Consequently, an isomorphism between two given well-ordered sets will be unique. #Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules. #Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface. #Rigidity results in K-theory show isomorphisms between various algebraic K-theory groups.
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The adjacency matrix of a signed graph Σ on n vertices is an n × n matrix A(Σ). It has a row and column for each vertex. The entry avw in row v and column w is the number of positive vw edges minus the number of negative vw edges. On the diagonal, avv = 0 if there are no loops or half-edges; the correct definition when such edges exist depends on the circumstances.
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The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with m edges is triangle-free in time O(m1.41). Another approach is to find the trace of A3, where A is the adjacency matrix of the graph.
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The change also saw the station move from AM radio to FM, broadcasting on 104.9 FM. The station was sold to Astral Media in 2007. St. Albert lost the radio station after its most recent change in 2011, when it moved to Edmonton after changing formats from adult contemporary to top 40, becoming 104.9 Virgin Radio. Due to the city's adjacency to Edmonton, all major Edmonton media—newspapers, television, and radio—also serve St. Albert.
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In written language there are frameworks that are used to portray this type of language. Discourse structure (which can also be found in spoken language) is how the conversation is organized, in which adjacency pairs - an utterance and the answer to that utterance - are used. Discourse markers are used to organize conversation ("first," "secondly," etc.). Lexis denotes the words being used in a text or spoken; these words can create a semantic field.
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Algona is a city in King County, Washington, United States, and the Seattle metropolitan area, surrounded by the suburbs of Auburn to the north and east, Pacific to the south, and unincorporated King County to the west. The population was 3,014 as of the 2010 census and an estimated 3,223 in 2018. Due to Algona's adjacency to the city of Pacific, the two communities are sometimes referred to collectively as Algona-Pacific or Algona/Pacific.
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Exetel has over 300Gbit/s of bandwidth linking its customers to Exetel and over 300Gbit/s+ of bandwidth linking Exetel to national and international internet networks. Multiple 10Gbit connections to numerous peering networks are used to ensure maximum content availability and network adjacency. Since it began operating, Exetel has implemented various practices to provide the maximum levels of on-net content to ensure the best possible end-user experience including Google, Akamai and Netflix caches.
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Efficiency: 4 pixels checked for each pixel filled (8 for an 8-way fill). A 4-way flood-fill algorithm that use the adjacency technique and a stack as its seed pixel store yields a linear fill with "gaps filled later" behaviour. This approach can be particularly seen in older 8-bit computer games, such as those created with Graphic Adventure Creator. Efficiency: 4 pixels checked for each pixel filled (8 for an 8-way fill).
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The city is named for its adjacency to Washington University. University City has much municipal open space, the largest parcels being Heman Park (which includes recreation and community centers and public pool facilities) and Ruth Park (a public golf course and nature trails). The city has four elementary schools, one middle school, two Catholic grade schools and one high school. The School District of University City is separately managed from other school districts in the area.
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It increases scalability up to 16 million logical networks and allows for layer 2 adjacency across IP networks. Multicast or unicast with head-end replication (HER) is used to flood broadcast, unknown unicast, and multicast (BUM) traffic. The VXLAN specification was originally created by VMware, Arista Networks and Cisco. Other backers of the VXLAN technology include Huawei, Broadcom, Citrix, Pica8, Big Switch Networks, Cumulus Networks, Dell EMC, Ericsson, Mellanox, FreeBSD, OpenBSD, Red Hat, Joyent, and Juniper Networks.
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If the nodes in the networks are not weighted, the structural components of the network can be shown through adjacency matrix. If the most elements in the matrix are zero, such matrix is referred as sparse matrix. In contrast, if most of the elements are nonzero, then the matrix is dense. The sparsity or density of the matrix is identified by the fraction of the zero element to the total number of the elements in the matrix.
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The location on Museum Mile was not seen as an advantageous adjacency. Slovin saw YIVO and the partner organizations as being more academic-focused. The Leo Baeck Institute was previously located at 129 East 73rd Street, and the American Jewish Historical Society, which had previously been a New York City-based institution prior to the early 1960s, was then located near Brandeis University in Waltham, Massachusetts. The Yeshiva University Museum was located in the Washington Heights neighborhood uptown.
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The date is set in accordance with the Hebrew calendar, on 27 Nisan, so that it varies in regard to the Gregorian calendar. Observance of the day is moved back to the Thursday before, if 27 Nisan falls on a Friday (as in 2021), or forward a day, if 27 Nisan falls on a Sunday (to avoid adjacency with the Jewish Sabbath, as in 2024). The fixed Jewish calendar ensures 27 Nisan does not fall on Saturday.
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Gunter has practiced medicine since 1996. Gunter works as an OB/GYN and a pain medicine physician. Her approach is based on evidence-based medicine integrated with a focus on empathy and the patient experience, which Gunter said she learned from the University of Western Ontario, and the adjacency to McMaster University Medical School, which is a center of evidence-based medicine. Since 2006, she has been at The Permanente Medical Group of Kaiser Permanente in Northern California.
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It forms the bulk of the retail commercial sector which is anchored by The Shops at Hilltop. This shopping center, like many around the country, killed downtown after opening in the 1970s. The area has become a booming bedroom community since a high increase in new housing being built in the desirable neighborhood. This area is of interest due to its proximity to job centers in Oakland and San Francisco, its relative adjacency compared to other suburbs, and affordability.
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A treemap showing the exports of Uganda Many data sets, such as spatial entities of countries or common structures for governments, can be organized into natural hierarchies. Node-link diagrams, adjacency diagrams, and enclosure diagrams are all types of infographics that effectively communicate hierarchical data. Node-link diagrams are a popular method due to the tidy and space-efficient results. A node-link diagram is similar to a tree, where each node branches off into multiple sub-sections.
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Colours indicate syntactic function; from lowest to highest mean trophic level: nouns (blue), prepositions and conjunctions (cyan), determiners (pink), adverbs (yellow), pronouns (green), verbs (red), and adjectives (purple). When a word has more than one function, the one most common in the text is used. The relation between trophic level and node function can be seen in networks other than food webs. The figure shows a word adjacency network derived from the book Green Eggs and Ham, by Dr Seuss.
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For each vertex we store the list of adjacencies (out-edges) in order of the planarity of the graph (for example, clockwise with respect to the graph's embedding). We then initialize a counter i = n + 1 and begin a Depth-First Traversal from s. During this traversal, the adjacency list of each vertex is visited from left-to-right as needed. As vertices are popped from the traversal's stack, they are labelled with the value i, and i is then decremented.
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Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like sociology. For example, eigenvector centrality uses the eigenvectors of the adjacency matrix corresponding to a network, to determine nodes that tend to be frequently visited. Formally established measures of centrality are degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, subgraph centrality and Katz centrality. The purpose or objective of analysis generally determines the type of centrality measure to be used.
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Hadjichristidis N., Pispas S., Floudas G. Block copolymers: synthetic strategies, physical properties, and applications – Wiley, 2003. An emerging technique is chain shuttling polymerization. The synthesis of block copolymers requires that both reactivity ratios are much larger than unity (r1 >> 1, r2 >> 1) under the reaction conditions, so that the terminal monomer unit of a growing chain tends to add a similar unit most of the time. The "blockiness" of a copolymer is a measure of the adjacency of comonomers vs their statistical distribution.
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The degree or valency of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice. In a graph of order , the maximum degree of each vertex is (or if loops are allowed), and the maximum number of edges is (or if loops are allowed). The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Specifically, two vertices and are adjacent if is an edge.
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Adding a name to the triple makes a "quad store" or named graph. A graph database has a more generalized structure than a triplestore, using graph structures with nodes, edges, and properties to represent and store data. Graph databases might provide index-free adjacency, meaning every element contains a direct pointer to its adjacent elements, and no index lookups are necessary. General graph databases that can store any graph are distinct from specialized graph databases such as triplestores and network databases.
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2004, Mississauga, Canada At the edge of the university’s campus, enclosed by a park and a courtyard garden, sits the CCIT building. The building is a place of occupancy, adjacency, and transition, as it creates a linear public circulation space between the Student Centre and the Library. The concept was to bring the natural landscape through the building, into the courtyard garden, and through to the campus. The fully transparent glazed façade diminishes the line between the interior and exterior.
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Queries using nested sets can be expected to be faster than queries using a stored procedure to traverse an adjacency list, and so are the faster option for databases which lack native recursive query constructs, such as MySQL. However, recursive SQL queries can be expected to perform comparably for 'find immediate descendants' queries, and much faster for other depth search queries, and so are the faster option for databases which provide them, such as PostgreSQL, Oracle, and Microsoft SQL Server.
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The traversal maps constitute a partial unary algebra that forms a basis for representing trees as linked data structures. At least conceptually,there are parent links, sibling adjacency links, and first / last child links. This also applies to unordered trees in general, which can be observed on the dentry data structure in the Linux VFS. Similarly to the "V-S-H-L±" system of partial orders, there are pairs of traversal maps that uniquely determine the whole ordered tree structure.
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An upper limit gives the network a global size and assures a fixed maximum response time for a network without any moving nodes. The algorithm has a few special features to cope with cases that are common in radio networks, such as unidirectional links, and looped-transmission caused by out-of-date routing tables. In particular, it reroutes all transmissions to nearby nodes whenever it loses a link to an adjacent node. It also retransmits its adjacency when this occurs.
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The Coxeter graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003 As a finite connected vertex- transitive graph that contains no Hamiltonian cycle, the Coxeter graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for an Hamiltonian path and is verified by the Coxeter graph.
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STEM currently includes a large number of plug-ins for the 244 countries and dependent areas defined by the Geographic Coding Standard maintained by the International Organization for Standardization. These plug-ins contain global data including geographic data, population data, demographics, and basic models of disease. The disease models distributed with STEM include epidemiological compartment models. Other plug-ins describe relationships between regions including nearest-neighbor or adjacency relationships as well as information about transportation, such as connections by roads and a model of air transportation.
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One of the dominant issues in the election campaign was the status of the city's St. Joseph's Hospital. With the completion of the new Sudbury Regional Hospital facility, the closure of the St. Joseph's site was imminent — however, despite the city's longtime interest in acquiring the property due to its adjacency to Bell Park and Lake Ramsey, the Sisters of St. Joseph sold the site in 2010 to Panoramic Properties, a condominium developer from Niagara Falls."St. Joseph's slipped though council's hands". Sudbury Star, June 2010.
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Metal complex dyes using copper or chromium are commonly used for producing dull colors. Tridentate ligand dyes are useful because they are more stable than their bi- or mono-dentate counterparts.Grychtol, K.; Mennicke, W. "Metal- Complex Dyes." In Ullmann’s Encyclopedia of Industrial Chemistry; 2002, Wiley- VCH, Hunger, K.; Mischke, P.; Rieper, W.; Raue, R.; Kunde, K.; Engel, A. "Azo Dyes." In Ullmann’s Encyclopedia of Industrial Chemistry, 2002, Wiley-VCH, :500px :350px Due to the adjacency of the amino and hydroxyl groups, 2-aminophenol readily forms heterocycles.
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An algebraic representation of the Hamiltonian cycles of a given weighted digraph (whose arcs are assigned weights from a certain ground field) is the Hamiltonian cycle polynomial of its weighted adjacency matrix defined as the sum of the products of the arc weights of the digraph's Hamiltonian cycles. This polynomial is not identically zero as a function in the arc weights if and only if the digraph is Hamiltonian. The relationship between the computational complexities of computing it and computing the permanent was shown in .
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If is the adjacency matrix of the directed or undirected graph , then the matrix (i.e., the matrix product of copies of ) has an interesting interpretation: the element gives the number of (directed or undirected) walks of length from vertex to vertex . If is the smallest nonnegative integer, such that for some , , the element of is positive, then is the distance between vertex and vertex . This implies, for example, that the number of triangles in an undirected graph is exactly the trace of divided by 6.
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W Architecture & Landscape Architecture (W Architecture) is an international architecture and landscape architecture firm based in Brooklyn, New York City. Founded in 1999 by Barbara E. Wilks, the firm is primarily known for its design of major waterfront reclamation projects and collaborative repurposing of public spaces. W Architecture has received substantial coverage in the media for the Edge Park in Williamsburg, Brooklyn; a redesign of the West Harlem waterfront; restoration of St. Patrick's Island in Calgary; and the recent Plaza 33 Madison Square Garden adjacency.
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Vuillemin invented the binomial heap and Cartesian tree data structures. With Ron Rivest, he proved the Aanderaa–Rosenberg conjecture, according to which any deterministic algorithm that tests a nontrivial monotone property of graphs, using queries that test whether pairs of vertices are adjacent, must perform a quadratic number of adjacency queries. In the 1980s, Vuillemin was the director of a project to develop a workstation using VLSI technology, under which the Le Lisp programming language was developed. With Franco P. Preparata, he also introduced the cube-connected cycles as a network topology in parallel computing.
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In photographic science, a Mackie line is an adjacency or border effect created during development, at the border between areas of high and low densities. During developing, developer remains relatively fresh in an area of low density as less developing takes place, and consequently, developer oxidation product concentration remains relatively low. At the border between high and low density areas the relatively fresh developer diffuses laterally into the high density area and causes there a continuation of development. The result is an increased border density of the high density area.
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Rantalainen, Timo (1995), "Location of Mobile Station in the GSM network", Master's thesis available at Aalto University Otaniemi Main Library (P1 Ark S80). In 1990 International Teletrac Systems (later PacTel Teletrac), founded in Los Angeles CA, introduced the world's first dynamic real-time stolen vehicle recovery services. As an adjacency to this they began developing location based services that could transmit information about location-based goods and services to custom-programmed alphanumeric Motorola pagers. In 1996 the US Federal Communications Commission (FCC) issued rules requiring all US mobile operators to locate emergency callers.
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A simple technical way to solve this problem is to extend the input graph to a complete bipartite graph, by adding artificial edges with very large weights. These weights should exceed the weights of all existing matchings, to prevent appearance of artificial edges in the possible solution. As shown by Mulmuley, Vazirani and Vazirani, the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph. Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least ½.
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In 211, Han Sui and Ma Chao drew up armies against Cao Cao in response to a suspected invasion of their lands. Cities throughout Hongnong and Fengyi commanderies rose in support, while Hedong Commandery remained firmly loyal, despite adjacency to the rebels. When Cao Cao marched west to confront them, his army was fed entirely by grain from Hedong, and after defeating Han Sui and Ma Chao there were still in excess of 4,000 hectolitres of grain left over.Loewe, 65 Cao Cao increased Du Ji's salary after this.
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In graph theory, a Sachs subgraph of a given graph is a subgraph in which all connected components are either single edges or cycles. These subgraphs are named after Horst Sachs, who used them in an expansion of the characteristic polynomial of the adjacency matrix of graphs. A similar expansion using Sachs subgraphs is also possible for permanental polynomials of graphs. Sachs subgraphs and the polynomials calculated with their aid have been applied in chemical graph theory, for instance as part of a test for the existence of non-bonding orbitals in hydrocarbon structures.
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Its adjacency to the palace may have been one reason why it did not feature a minaret; in order to prevent anyone climbing the minaret from looking down into the caliph's palaces. The Mamluk amir Yalbugha al-Salimi restored the mosque in 1393 or 1397 and added a minaret (which collapsed in 1412 and was later restored) as well as shop stalls to the right of the entrance. Al-Salimi also restored or replaced the minbar, the mihrab, and the ablutions area. In 1993, the mosque was extensively and heavily renovated by the Dawoodi Bohras.
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The main idea for their algorithm is to generate unions of subsets of incrementally (for each set in , forming its union with all previously generated unions), represent the resulting family of sets in a trie, and use the trie representation to test certain candidate pairs of sets for adjacency in the covering relation; it takes time . In later work, the same authors showed that the algorithm could be made fully incremental (capable of adding elements to the partial order one at a time) with the same total time bound..
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The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory. For example, if A is the adjacency matrix of a n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA.. As a second example, the matrix appears in some linear-algebraic proofs of Cayley's formula, which gives the number of spanning trees of a complete graph, using the matrix tree theorem.
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Because of its adjacency to this railway, descriptions of the history of the Dunes often mention Tremont. For example, the Prairie Club beach house built in 1913 by landscape preservationist Jens Jensen and a group of friends was built in Tremont, close to the rail line."Reading 3: Beauty of the Wild", U.S. National Park Service, accessed November 21, 2008. In 1929, "scarcely a building of New City West survived"; however, new buildings on the site of the old city led to its repopulation as the summer resort town of Tremont.
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An important set of NEXPTIME-complete problems relates to succinct circuits. Succinct circuits are simple machines used to describe graphs in exponentially less space. They accept two vertex numbers as input and output whether there is an edge between them. If solving a problem on a graph in a natural representation, such as an adjacency matrix, is NP-complete, then solving the same problem on a succinct circuit representation is NEXPTIME-complete, because the input is exponentially smaller (under some mild condition that the NP-completeness reduction is achieved by a "projection").
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Considering the loose soil of Shanghai, its adjacency to high building on three sides, its closeness to the Bund and the din produced by cars on the street, the difficulties of digging are evident. Although the foundation has not been completed for now, we can tell how grand the construction plan is by simply judging from the completed part. It is told that the outlook of the building will convey a sense of magnificence through a simple style. The surrounding wall in the lower part will use granites from Suzhou.
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In spectral graph theory, the Alon–Boppana bound provides a lower bound on the second-largest eigenvalue of the adjacency matrix of a d-regular graph, meaning a graph in which every vertex has degree d. The reason for the interest in the second-largest eigenvalue is that the largest eigenvalue is guaranteed to be d due to d-regularity, with the all-ones vector being the associated eigenvector. The graphs that come close to meeting this bound are Ramanujan graphs, which are examples of the best possible expander graphs.
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The confrontation in which Julio was killed occurred in the Belize-Guatemala adjacency zone, on the Belizean side, 562 meters from the border, on the 20 April 2016. According to Belizean authorities, Belize Defence Force personnel and FCD park rangers were resting by a corn field when at 7 pm Julio's father, Carlos Ramirez, and other companions started flashing lights. Upon being asked to stop, Ramirez and his associates began shooting at the personnel, prompting the personnel to return fire. During the confrontation, Julio's father and brother were injured but escaped to Guatemala.
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Every two-graph is equivalent to a set of lines in some dimensional euclidean space each pair of which meet in the same angle. The set of lines constructed from a two graph on n vertices is obtained as follows. Let -ρ be the smallest eigenvalue of the Seidel adjacency matrix, A, of the two-graph, and suppose that it has multiplicity n - d. Then the matrix is positive semi-definite of rank d and thus can be represented as the Gram matrix of the inner products of n vectors in euclidean d-space.
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The role of central conceptual structures in the development of children's thought. Monographs of the Society for Research in Child Development, 61 (1-2, Serial No. 246). Case and his colleagues identified central conceptual structures for quantities, space, social behavior, narrative, music, and motor behavior. Each of these structures is supposed to involve a set of core processes and principles which serve to organize a broad array of situations; for example, the concept of more and less for quantities, adjacency and inclusion relationships for space, and actions and intentions for social behavior.
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The 600 vertices of a 120-cell with an edge length of = 3− and a center-to-vertex radius of = 2 include all permutations of: : (0, 0, ±2, ±2) : (±1, ±1, ±1, ±) : (±φ−2, ±φ, ±φ, ±φ) : (±φ−1, ±φ−1, ±φ−1, ±φ2) and all even permutations of : (0, ±φ−2, ±1, ±φ2) : (0, ±φ−1, ±φ, ±) : (±φ−1, ±1, ±φ, ±2) where φ (also called τ) is the golden ratio, . Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter is 15, connecting each vertex to its coordinate-negation, at a Euclidean distance of 4 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from 2−3φ, with a multiplicity of 4, to 4, with a multiplicity of 1.
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64, 016132 (2001) In order to account for this so-called saturation effect, it has been proposed to weight edges inversely according to the number of common affiliations in the neighboring set. This is most easily achieved by introducing a scaling factor 1/(n - 1) onto the simple count, which weakens the link between nodes with more popular common matches. # Weighting based on resource allocation. With simple and hyperbolic weighting, the projected adjacency matrix is always set to be symmetrical, which implies that a link between two projected nodes carries the same weight for both vertices.
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Ninh Bình () is a province of Vietnam, in the Red River Delta region of the northern part of the country. The province is famous for a high density of natural and cultural attractions, including reserved parks in Cúc Phương National Park and Vân Long, grotto caves and rivers in Tràng An, Tam Cốc-Bích Động and Múa Caves, historic monuments in the Hoa Lư ancient capital, Vietnam's largest buddhist worshiping complex (Bái Đính Temple), and the Phát Diệm Cathedral with "eclectic architectural style". Thanks to its adjacency to Hanoi, day trips from the capital are easily manageable.
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Subsequently, Frieze and Kannan gave a different version and extended it to hypergraphs. They later produced a different construction due to Alan Frieze and Ravi Kannan that uses singular values of matrices. One can find more efficient non- deterministic algorithms, as formally detailed in Terence Tao's blog and implicitly mentioned in various papers. An inequality of Terence Tao extends the Szemerédi regularity lemma, by revisiting it from the perspective of probability theory and information theory instead of graph theory.. Terence Tao has also provided a proof of the lemma based on spectral theory, using the adjacency matrices of graphs.
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South Pasadena also has a strong claim to having the oldest and most historic sites in the San Gabriel Valley. For many centuries, its adjacency to a natural fording place along the Arroyo Seco had served as a gateway to travel and commerce for aboriginal peoples here and along the coast. It was here that Hahamognas greeted Portola and the missionaries who later established the San Gabriel Mission a few miles to the east. The initial buildings on the Rancho San Pascual were built on the land which eventually became the cities of Pasadena, South Pasadena and Altadena.
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Discussing the album's title, People and Things, McMahon said he loved the starkness of the two words' adjacency, which he has said references the tone of the songs and the transparency of love from a different viewpoint. McMahon thought of the title while working on the band's seventh EP Dear Jack EP (2009) in a studio with Dean. He viewed the album's release as the final part of the Jack's Mannequin story, which was initially intended to only last for one album. Everything in Transit deals with McMahon's break-up with his girlfriend, whom he would eventually marry.
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Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix. Let be an integer that is relatively prime to , and let be any integer. Then the linear function that takes a number to transforms a circulant numbering to another circulant numbering.
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On November 18, 2009, the BSA announced that it had chosen the West Virginia site, known locally as the Garden Ground Mountain property, as the future home of the Summit Bechtel Reserve. One of the deciding factors for Project Arrow in choosing the West Virginia site was its adjacency to New River Gorge National River. More than 13 miles of the property borders the park, giving Scouts and Scouters access to the more than 70,000 acres of managed wilderness beyond the Reserve's property. On April 1, 2012, Justin D. (Dan) McCarthy was appointed Summit Group Director.
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Many of the exceptional objects in mathematics and physics have been found to be connected to each other. Developments such as the Monstrous moonshine conjectures show how, for example, the Monster group is connected to string theory. The theory of modular forms shows how the algebra E8 is connected to the Monster group. (In fact, well before the proof of the Monstrous moonshine conjecture, the elliptic j-function was discovered to encode the representations of E8.) Other interesting connections include how the Leech lattice is connected via the Golay code to the adjacency matrix of the dodecahedron (another exceptional object).
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PAETEC Headquarters was a building proposed and approved for construction in Rochester, New York. It was to serve as the new headquarters for the Rochester-based telecommunications company, PAETEC Holding Corp. Its original proposed height was 40 floors, with a LEED-certified rooftop garden and was originally slated for completion in 2012. The proposed PAETEC building would have sat on the southeast corner of the intersection of Main Street and Clinton Avenue, taking advantage of both the refreshed (abated) skeletal structure of the former Seneca Building and adjacency to an enormous underground parking facility (1600+spaces) and truck tunnel.
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In contrast, this equation has no solution in the finite fields Z/p where p is an odd prime but is not Pythagorean., p. 100. The Paley graph with 13 vertices For every Pythagorean prime p, there exists a Paley graph with p vertices, representing the numbers modulo p, with two numbers adjacent in the graph if and only if their difference is a quadratic residue. This definition produces the same adjacency relation regardless of the order in which the two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue..
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Let G=(V,E,w) be a directed Graph with the set of nodes V and the set of edges E\subseteq V\times V. Each edge e \in E has a weight w(e) assigned. The goal of the all-pair-shortest-paths problem is to find the shortest path between all pairs of nodes of the graph. For this path to be unique it is required that the graph does not contain cycles with a negative weight. In the remainder of the article it is assumed that the graph is represented using an adjacency matrix.
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Distinguishing coloring of a 4-hypercube graph In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph.
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Beside Sneferka's serekh, the inscriptions mention several institutions and places already known thanks to finds dating to Qa'a's reign. They are called Qau-Netjeru ("Elevations of the gods") and Ah-Netjer ("Divine palace") and appear in several stone vessel inscriptions from Qa'a's tomb at Abydos. Egyptologists such as Peter Kaplony conclude that the inscriptions prove a chronological adjacency to king Qa'a or that the name "Sneferka" was an alternative name that Qa'a bore for a short time. Two artifacts of different origins show the serekh of a king, whose name is highly disputed, for the hieroglyphic sign used to write the king's name is almost illegible.
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Using this method, planar graphs and many more restricted families of graphs may be encoded using a number of bits that is information- theoretically optimal: if there are Pn n-vertex graphs in the family of graphs to be represented, then an individual graph in the family can be represented using only (1 + o(n))log2Pn bits.. It is also possible to construct representations of this type in which one may test adjacency between vertices, determine the degree of a vertex, and list neighbors of vertices in constant time per query, by augmenting the table of subgraphs with additional tabular information representing the answers to the queries...
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NodeXL Pro imports UCINet and GraphML files, as well as Excel spreadsheets containing edge lists or adjacency matrices, into NodeXL workbooks. NodeXL Pro also allows for quick collection of social media data via a set of import tools which can collect network data from e-mail, Twitter, YouTube, and Flickr. NodeXL requests the user's permission before collecting any personal data and focuses on the collection of publicly available data, such as Twitter statuses and follows relationships for users who have made their accounts public. These features allow NodeXL users to instantly get working on relevant social media data and integrate aspects of social media data collection and analysis into one tool.
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Perle (1996), p.22. Cognate set created from paired interval-7 cycles of sum 0 0 7 2 9 4 11 6 1 8 3 10 5 (0 \+ 0 5 10 3 8 1 6 11 4 9 2 7 (0 ________________________________________ = 0 0 0 0 0 0 0 0 0 0 0 0 (0 The two cycles may also be aligned as pairs of sum 7 or sum 5 dyads. All together these pairs of cycles form a set complex, "any cyclic set of the set complex may be uniquely identified by its two adjacency sums," and as such the example above shows p0p7 and i5i0.Perle (1996), p.23.
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Another application of this concept lies in sparse matrix computations, namely for using nested dissection to compute the Cholesky factorization of a (symmetric) matrix in parallel. A given sparse (n\times n)-matrix M may be interpreted as the adjacency matrix of some symmetric digraph G on n vertices, in a way such that the non-zero entries of the matrix are in one-to-one correspondence with the edges of G. If the cycle rank of the digraph G is at most k, then the Cholesky factorization of M can be computed in at most k steps on a parallel computer with n processors .
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Lila Genus Martinez was born on 21 October 1950 near Punta Gorda in a disputed border region, now known as the "adjacency zone", between British Honduras and Guatemala. Neither country wanted to issue a birth certificate, but eventually, her parents Gladys Matilda Genus and Alfredo Martinez Jiménez were able to secure a birth certificate and passport from Guatemala. Her father was a chiclero, traveling in chicle camps along the border to gather the natural gum and her mother was a cook. Her father was descended of Jemima Hariniah Hause, a runaway slave originally from Sierra Leone, who fled Mexico to Belize and married a Belizean named Harrison.
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In general, the parallel edge processing based on 2D partitioning can be organized in 2 communication phases, which are "expand" phase and "fold" phase. In the "expand" phase, if the edge list for a given vertex is the column of the adjacency matrix, then for each vertex v in the frontier, the owner of v is responsible to tell other processors in its processor column that v is visited. That's because each processor only stores partial edge lists of vertexes. After this communication, each processor can traverse the column of according to the vertexes and find out their neighbors to form the next frontier.
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In mathematics, a Perron number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than α in absolute value. For example, the larger of the two roots of the irreducible polynomial x^{2} -3x + 1 is a Perron number. Perron numbers are named after Oskar Perron; the Perron–Frobenius theorem asserts that, for a real square matrix with positive algebraic coefficients whose largest eigenvalue is greater than one, this eigenvalue is a Perron number. As a closely related case, the Perron number of a graph is defined to be the spectral radius of its adjacency matrix.
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In mathematics, the Coates graph or Coates flow graph, named after C.L. Coates, is a graph associated with the Coates' method for the solution of a system of linear equations. The Coates graph Gc(A) associated with an n × n matrix A is an n-node, weighted, labeled, directed graph. The nodes, labeled 1 through n, are each associated with the corresponding row/column of A. If entry aji ≠ 0 then there is a directed edge from node i to node j with weight aji. In other words, the Coates graph for matrix A is the one whose adjacency matrix is the transpose of A.
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The sum of weighted perfect matchings can also be computed by using the Tutte matrix for the adjacency matrix in the last step. Kuratowski's theorem states that : a finite graph is planar if and only if it contains no subgraph homeomorphic to K5 (complete graph on five vertices) or K3,3 (complete bipartite graph on two partitions of size three). Vijay Vazirani generalized the FKT algorithm to graphs that do not contain a subgraph homeomorphic to K3,3. Since counting the number of perfect matchings in a general graph is #P-complete, some restriction on the input graph is required unless FP, the function version of P, is equal to #P.
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A zero in these locations will be incorrectly interpreted as an edge with no distance, cost, etc. If W is an n \times n matrix containing the edge weights of a graph, then W^k (using this distance product) gives the distances between vertices using paths of length at most k edges, and W^n is the distance matrix of the graph. An arbitrary graph on vertices can be modeled as a weighted complete graph on vertices by assigning a weight of one to each edge of the complete graph that corresponds to an edge of and zero to all other edges. for this complete graph is the adjacency matrix of .
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In the monadic second-order logic of graphs, the variables represent objects of up to four types: vertices, edges, sets of vertices, and sets of edges. There are two main variations of monadic second-order graph logic: MSO1 in which only vertex and vertex set variables are allowed, and MSO2 in which all four types of variables are allowed. The predicates on these variables include equality testing, membership testing, and either vertex-edge incidence (if both vertex and edge variables are allowed) or adjacency between pairs of vertices (if only vertex variables are allowed). Additional variations in the definition allow additional predicates such as modular counting predicates.
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During World War II, uranium was mined on the Diné and Hopi reservations. The dangers of radiation exposure were not adequately explained to the native people, who made up almost all the workforce of these mines, and lived in their immediate adjacency. As a result, some residents who lived near the uranium projects used the quarried rock from the mines to build their houses, these materials were radioactive and had detrimental health effects on the residents, including increased rates of kidney failure and cancer. During extraction some native children would play in large water pools which were heavily contaminated with uranium created by mining activities.
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Switching equivalence means that two graphs are related by switching, and an equivalence class of signed graphs under switching is called a switching class. Sometimes these terms are applied to equivalence of signed graphs under the combination of switching and isomorphism, especially when the graphs are unlabeled; but to distinguish the two concepts the combined equivalence may be called switching isomorphism and an equivalence class under switching isomorphism may be called a switching isomorphism class. Switching a set of vertices affects the adjacency matrix by negating the rows and columns of the switched vertices. It affects the incidence matrix by negating the rows of the switched vertices.
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Among abelian Cayley graphs that are strongly regular and have the last two parameters differing by one, it is the only graph that is not a Paley graph. It is also an integral graph, meaning that the eigenvalues of its adjacency matrix are integers. Like the 9\times 9 Sudoku graph it is an integral abelian Cayley graph whose group elements all have order 3, one of a small number of possibilities for the orders in such a graph. There are five possible combinations of parameters for strongly regular graphs that have one shared neighbor per pair of adjacent vertices and exactly two shared neighbors per pair of non-adjacent vertices.
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The trace is zero if and only if the graph is triangle-free. For dense graphs, it is more efficient to use this simple algorithm which relies on matrix multiplication, since it gets the time complexity down to O(n2.373), where n is the number of vertices. As showed, triangle-free graph recognition is equivalent in complexity to median graph recognition; however, the current best algorithms for median graph recognition use triangle detection as a subroutine rather than vice versa. The decision tree complexity or query complexity of the problem, where the queries are to an oracle which stores the adjacency matrix of a graph, is Θ(n2).
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This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations. In mathematics, juxtaposition of symbols is the adjacency of factors with the absence of an explicit operator in an expression, especially for commonly used for multiplication: ax denotes the product of a with x, or a times x. It is also used for scalar multiplication, matrix multiplication, and function composition. In numeral systems, juxtaposition of digits has a specific meaning.
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"Geographically isolated wetlands are important biogeochemical reactors to the landscape", an article Basu co-authored, gains relevance in a time in which the Trump administration is rolling back clean water protections set in place during the Obama-era. These protections previously placed limits on chemicals that were allowed to be used near streams, wetlands, and other bodies of water. Without these protections a permit will no longer be necessary to dispose of potentially harmful chemicals into streams and wetlands. The authors concluded that despite the lack of connection to surface water or adjacency to jurisdictional waters that other wetlands have, geographically isolated wetlands are essential to biogeochemical processing of landscapes.
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In Distributed Morphology, the linear order of morphemes is determined by their hierarchical position in the syntactic structure, as well as by certain post-syntactic operations. Head movement is the main syntactic operation determining morpheme order, while Morphological Merger (or Merger under Adjacency) is the main post-syntactic operation targeting affix order. Other post-syntactic operations that might affect morpheme order are Lowering and Local Dislocation (see previous section for details on these operations). The general principle behind morpheme order is the Mirror Principle (first formulated by Baker 1985), according to which the linear order of morphemes is the mirror image of the hierarchy of syntactic projections.
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Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index. As a result, the topological index can be defined as two-dimensional descriptors that can be easily calculated from the molecular graphs, and do not depend on the way the graph is depicted or labeled and no need of energy minimization of the chemical structure.
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Here "sub-exponential time" is taken to mean the second definition presented below. (On the other hand, many graph problems represented in the natural way by adjacency matrices are solvable in subexponential time simply because the size of the input is square of the number of vertices.) This conjecture (for the k-SAT problem) is known as the exponential time hypothesis. Since it is conjectured that NP-complete problems do not have quasi-polynomial time algorithms, some inapproximability results in the field of approximation algorithms make the assumption that NP-complete problems do not have quasi-polynomial time algorithms. For example, see the known inapproximability results for the set cover problem.
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Map of Winstanley Estate, Battersea Winstanley Estate and York Road Estate are large estates of predominantly public housing apartments in Battersea, London, adjacent to Clapham Junction railway station, although some have since passed into private ownership. Due to their proximity to one another, the Winstanley Estate and York Road Estate have historically been grouped together and shared facilities, including York Gardens and transport links at Clapham Junction. According to official data, there are a total of 1419 homes on the estates, with approximately 5200 residents. The adjacency of a built-up residential area to one of the busiest stations in Europe is almost unique and has profoundly influenced various factors of the district.
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Since the very opening, due to its adjacency to the Budapest University Library, it served as a cultural nexus and an agreed upon meeting place among multiple parties from the university and the other cultural elite of the time. It was common to see advertisements announcing meeting and discussions to take place at the Central Cafe, as well as for political rallies and even politically motivated fist fights to take place at the Central Cafe and the adjacent square. Many civil and political organization were formed and were operating from the cafe. By the end of 1918, the Hungarian Socialist Democratic party was found and set up its party headquarters in one of the Central's rooms.
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Data restructuring can be performed by a GIS to convert data into different formats. For example, a GIS may be used to convert a satellite image map to a vector structure by generating lines around all cells with the same classification, while determining the cell spatial relationships, such as adjacency or inclusion. More advanced data processing can occur with image processing, a technique developed in the late 1960s by NASA and the private sector to provide contrast enhancement, false color rendering and a variety of other techniques including use of two dimensional Fourier transforms. Since digital data is collected and stored in various ways, the two data sources may not be entirely compatible.
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If C is a symmetric conference matrix of order n > 1, then not only must n be congruent to 2 (mod 4) but also n − 1 must be a sum of two square integers;Belevitch, p.240 there is a clever proof by elementary matrix theory in van Lint and Seidel. n will always be the sum of two squares if n − 1 is a prime power.Stinson, p.78 Given a symmetric conference matrix, the matrix S can be viewed as the Seidel adjacency matrix of a graph. The graph has n − 1 vertices, corresponding to the rows and columns of S, and two vertices are adjacent if the corresponding entry in S is negative.
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Granular computing (GrC) is an emerging computing paradigm of information processing that concerns the processing of complex information entities called "information granules", which arise in the process of data abstraction and derivation of knowledge from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like. At present, granular computing is more a theoretical perspective than a coherent set of methods or principles. As a theoretical perspective, it encourages an approach to data that recognizes and exploits the knowledge present in data at various levels of resolution or scales.
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In general, a distance matrix is a weighted adjacency matrix of some graph. In a network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes.Frank Harary, Robert Z. Norman and Dorwin Cartwright (1965) Structural Models: An Introduction to the Theory of Directed Graphs, pages 134–8, John Wiley & Sons This distance function, while well defined, is not a metric. There need be no restrictions on the weights other than the need to be able to combine and compare them, so negative weights are used in some applications.
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For some graphs, such as bipartite graphs and high- degree planar graphs, the number of colors is always , and for multigraphs, the number of colors may be as large as . There are polynomial time algorithms that construct optimal colorings of bipartite graphs, and colorings of non- bipartite simple graphs that use at most colors; however, the general problem of finding an optimal edge coloring is NP-hard and the fastest known algorithms for it take exponential time. Many variations of the edge coloring problem, in which an assignments of colors to edges must satisfy other conditions than non-adjacency, have been studied. Edge colorings have applications in scheduling problems and in frequency assignment for fiber optic networks.
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A sample DSM with 7 elements and 11 dependency marks. The design structure matrix (DSM; also referred to as dependency structure matrix, dependency structure method, dependency source matrix, problem solving matrix (PSM), incidence matrix, N2 matrix, interaction matrix, dependency map or design precedence matrix) is a simple, compact and visual representation of a system or project in the form of a square matrix.S.D. Eppinger and T.R. Browning, Design Structure Matrix Methods and Applications, MIT Press, Cambridge, 2012. It is the equivalent of an adjacency matrix in graph theory, and is used in systems engineering and project management to model the structure of complex systems or processes, in order to perform system analysis, project planning and organization design.
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Red then places a marker on any unoccupied space; black does the same; and so on until all eight markers are on the board. The object of the game is for either player to win by having all four of his markers in a straight line (vertical, horizontal, or diagonal) or on a square of four adjacent spaces. (Adjacency is horizontal, vertical, or diagonal, but does not wrap around the edges of the board.) If neither player has won after the "drop" (when all eight pieces are on the board), then they move their pieces one at a time, with Black playing first. A piece may be moved only to an adjacent space.
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Small modifications and an added depth-first search step produce canonical labeling of such uniformly-chosen random graphs in linear expected time. This result sheds some light on the fact why many reported graph isomorphism algorithms behave well in practice... This was an important breakthrough in probabilistic complexity theory which became widely known in its manuscript form and which was still cited as an "unpublished manuscript" long after it was reported at a symposium. A commonly known canonical form is the lexicographically smallest graph within the isomorphism class, which is the graph of the class with lexicographically smallest adjacency matrix considered as a linear string. However, the computation of the lexicographically smallest graph is NP-hard.
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The design of the future interchange in west Whitecourt is anticipated to be complicated by the convergence of two existing highways, the adjacency of the CN rail line, the short distance between Highway 32 and the McLeod River to the south, existing highway commercial and industrial development along Highway 43 and Highway 32 respectively, and the close proximity of another interchange opportunity to the southeast at Govenlock Road/West Mountain Road, just east of the McLeod River. The right-of-way is designated as Highway 43X. A functional planning study was initiated in 2008 for the Highway 43/43X interchange southeast of Whitecourt. The study considers three options, all of which incorporate a direct connection to East Mountain Road.
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Luboszyce () is a village in the administrative district of Gmina Gubin, within Krosno Odrzańskie County, Lubusz Voivodeship, in western Poland, close to the German border. It lies approximately south of Gubin, south-west of Krosno Odrzańskie, and west of Zielona Góra. In the village there is a neo- gothic castle from 1846–1850 with an interesting landscape park,Krzysztof Garbacz: Przewodnik po zabytkach województwa lubuskiego (Guidebook for the monuments of Lubusz Voivodship), Agencja Wydawnicza PDN, Zielona Góra 2013, (Polish) while in adjacency – an important archaeological site of Luboszyce culture from 2nd-4th centuries (remains of a village and a cemetery).Grzegorz Domański: Kultura Luboszycka miedzy Łabą a Odrą w 2.–4. wieku („Luboszyce Culture between Elbe and Oder rivers in 2–4 centuries“).
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The village was once home to two sizeable public houses, The Mostyn Arms and The Queen's Head. These buildings flanked the church to the north and south, a proximity which was to be their undoing when the pious Lady Augusta Mostyn ordered their demolition in the latter years of the 19th century. The adjacency of the public houses to her estate was another factor which hastened their demise, as Lady Mostyn felt the nearness of such temptations was hampering the productivity of her workers. In 1898 Lady Mostyn came to an arrangement with the owner of the Mostyn arms - Sam Hughes - providing him with a freehold a short distance to the north on which he could build a new premises.
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A continuous-time quantum walk (CTQW) is a quantum walk on a given (simple) graph that is dictated by a time-varying unitary matrix that relies on the Hamiltonian of the quantum system and the adjacency matrix. The concept of a CTQW is believed to have been first considered for quantum computation by Edward Farhi and Sam Gutmann; since many classical algorithms are based on (classical) random walks, the concept of CTQWs were originally considered to see if there could be quantum analogues of these algorithms with e.g. better time-complexity than their classical counterparts. In recent times, problems such as deciding what graphs admit properties such as perfect state transfer with respect to their CTQWs have been of particular interest.
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However, its length and its various textual problems not characteristic of the riddles have led few scholars to pursue a simple riddle interpretation in modern textual study, and few such explanations have garnered serious attention in the recent history of its scholarship. Rather, the thematic similarity of the poem to The Wife's Lament, also found in the Exeter Book, has caused most modern scholars to place it, along with the Wife's Lament, solidly within the genre of the frauenlied, or woman's song and, more broadly, in that of the Old English elegy. These two poems are also used as examples of the female voice in broadening early feminist literary history. However, Wulf and Eadwacer's adjacency to the riddles has continued to inform commentary and interpretation.
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Beyond the Core argues that businesses must grow to survive—but only one in five strategies for growth succeeds. In the previous book, Profit from the Core, Zook explains how to grow profitably by concentrating on and harvesting the full potential in the core business. But if the core business fails to provide new growth, managers should stick to an expansion strategy based on various combinations of adjacency moves into areas away from, but related to, the core business, such as new product lines or new channels of distribution. This strategy is less risky than diversification; however, it can still create a substantial competitive advantage because the growth is based on or relates to the pack of skills that the business already does best.
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Due to its central location in Houston, Afton Oaks is in close proximity to several neighborhoods including Oak Estates, River Oaks, and Royden Oaks to the northeast, Lynn Park to the east, West University Place, Texas to the southeast, Bellaire, Texas to the south, and Tanglewood to the northwest. In addition to the Galleria, Highland Village, and River Oaks District shopping areas, Afton Oaks also enjoys adjacency to the Greenway Plaza business district. The major thoroughfares in Afton Oaks are Kettering Drive, Newcastle Drive, Richmond Avenue, and West Alabama Street. The notable signature elements of Afton Oaks include the esplanade along Newcastle Drive, the canopy of large, older hardwood trees throughout the neighborhood, a collection of rectangular column street markers, and a set of smaller triangular parks.
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RETAIN/370 ran special applications designed for technical support center use. Its most powerful feature was a full-text search engine, enabling most text documents in the system to be retrieved by using boolean search requests, similar in concept to full-text search engines in use today on the Internet, such as Google or AltaVista, although limited only to searching for individual words, or combinations of words, without reference to word-adjacency. RETAIN/370 was the first IBM system deployed on a large scale that had such a capability. The search engine component of RETAIN is called IRIS, for Interpretive Retrieval Information System (not to be confused with other non- IBM software systems of that name... IBM never sold this search engine as a product, so there was no trademark issue).
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If resource acquisition can fail without throwing an exception, such as by returning a form of `null`, it must also be checked before release, such as: f = open(filename) try: ... finally: if f: f.close() While this ensures correct resource management, it fails to provide adjacency or encapsulation. In many languages there are mechanisms that provide encapsulation, such as the `with` statement in Python: with open(filename) as f: ... The above techniques – unwind protection (`finally`) and some form of encapsulation – are the most common approach to resource management, found in various forms in C#, Common Lisp, Java, Python, Ruby, Scheme, and Smalltalk, among others; they date to the late 1970s in the NIL dialect of Lisp; see . There are many variations in the implementation, and there are also significantly different approaches.
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It is straightforward to verify that a given graph with n vertices and m edges is claw-free in time O(n4), by testing each 4-tuple of vertices to determine whether they induce a claw., p. 132. With more efficiency, and greater complication, one can test whether a graph is claw-free by checking, for each vertex of the graph, that the complement graph of its neighbors does not contain a triangle. A graph contains a triangle if and only if the cube of its adjacency matrix contains a nonzero diagonal element, so finding a triangle may be performed in the same asymptotic time bound as n × n matrix multiplication.. Therefore, using the Coppersmith–Winograd algorithm, the total time for this claw-free recognition algorithm would be O(n3.376).
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Oriented graph with corresponding incidence matrix An arrow is considered to be directed from x to y; y is called the head and x is called the tail of the arrow; y is said to be a direct successor of x and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arrow is called the inverted arrow of . The adjacency matrix of a multidigraph with loops is the integer-valued matrix with rows and columns corresponding to the vertices, where a nondiagonal entry aij is the number of arrows from vertex i to vertex j, and the diagonal entry aii is the number of loops at vertex i.
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The first part introduces the subject visually, encouraging the reader to think about packings not just as static objects but as dynamic systems of circles that change in predictable ways when the conditions under which they are formed (their patterns of adjacency) change. The second part concerns the proof of the circle packing theorem itself, and of the associated rigidity theorem: every maximal planar graph can be associated with a circle packing that is unique up to Möbius transformations of the plane. More generally the same result holds for any triangulated manifold, with a circle packing on a topologically equivalent Riemann surface that is unique up to conformal equivalence. The third part of the book concerns the degrees of freedom that arise when the pattern of adjacencies is not fully triangulated (it is a planar graph, but not a maximal planar graph).
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In a general sense, transition metal-based catalysis can be viewed as template reactions: Reactants coordinate to adjacent sites on the metal ion and, owing to their adjacency, the two reactants interconnect (insert or couple) either directly or via the action of another reagent. In the area of homogeneous catalysis, the cyclo- oligomerization of acetylene to cyclooctatetraene at a nickel(II) centre reflects the templating effect of the nickel, where it is supposed that four acetylene molecules occupy four sites around the metal and react simultaneously to give the product. This simplistic mechanistic hypotheses was influential in the development of these catalytic reactions. For example, if a competing ligand such as triphenylphosphine were added to occupy one coordination site, then only three molecules of acetylene could bind, and these come together to form benzene (see Reppe chemistry).
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The original application of chordal completion described in Computers and Intractability involves Gaussian elimination for sparse matrices. During the process of Gaussian elimination, one wishes to minimize fill-in, coefficients of the matrix that were initially zero but later become nonzero, because the need to calculate the values of these coefficients slows down the algorithm. The pattern of nonzeros in a sparse symmetric matrix can be described by an undirected graph (having the matrix as its adjacency matrix); the pattern of nonzeros in the filled-in matrix is always a chordal graph, any minimal chordal completion corresponds to a fill-in pattern in this way. If a chordal completion of a graph is given, a sequence of steps in which to perform Gaussian elimination to achieve this fill-in pattern can be found by computing an elimination ordering of the resulting chordal graph.
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Parts of the body removed in Whipple's operation Surgery with the intention of a cure is only possible in around one-fifth (20%) of new cases. Although CT scans help, in practice it can be difficult to determine whether the tumor can be fully removed (its "resectability"), and it may only become apparent during surgery that it is not possible to successfully remove the tumor without damaging other vital tissues. Whether or not surgical resection can be offered depends on various factors, including the precise extent of local anatomical adjacency to, or involvement of, the venous or arterial blood vessels, as well as surgical expertise and a careful consideration of projected post-operative recovery. The age of the person is not in itself a reason not to operate, but their general performance status needs to be adequate for a major operation.
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For example, if the input is an integer specified using unary coding, trial division will require only Θ(n1/2) arithmetic operations; but if the same input is specified in binary (or any larger base) the complexity rises to Θ(2n/2) operations, not because the algorithm is taking any additional time, but because the number of bits in the input n has become exponentially smaller. In the other direction, succinct circuits are compact representations of a limited class of graphs that occupy exponentially less space than ordinary representations like adjacency lists. Many graph algorithms on succinct circuits are EXPTIME-complete, whereas the same problems expressed with conventional representations are only P-complete, because the succinct circuit inputs have smaller encodings. Output-sensitive algorithms define their complexity not only in terms of their input but also their output.
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If edges (to child nodes) are thought of as references, then a tree is a special case of a digraph, and the tree data structure can be generalized to represent directed graphs by removing the constraints that a node may have at most one parent, and that no cycles are allowed. Edges are still abstractly considered as pairs of nodes, however, the terms and are usually replaced by different terminology (for example, and ). Different implementation strategies exist: a digraph can be represented by the same local data structure as a tree (node with value and list of children), assuming that "list of children" is a list of references, or globally by such structures as adjacency lists. In graph theory, a tree is a connected acyclic graph; unless stated otherwise, in graph theory trees and graphs are assumed undirected.
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Any square matrix A = (a_{ij}) can be viewed as the adjacency matrix of a directed graph, with a_{ij} representing the weight of the edge from vertex i to vertex j. Then, the permanent of A is equal to the sum of the weights of all cycle-covers of the graph; this is a graph-theoretic interpretation of the permanent. #SAT, a function problem related to the Boolean satisfiability problem, is the problem of counting the number of satisfying assignments of a given Boolean formula. It is a #P-complete problem (by definition), as any NP machine can be encoded into a Boolean formula by a process similar to that in Cook's theorem, such that the number of satisfying assignments of the Boolean formula is equal to the number of accepting paths of the NP machine.
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Any formula in SAT can be rewritten as a formula in 3-CNF form preserving the number of satisfying assignments, and so #SAT and #3SAT are equivalent and #3SAT is #P-complete as well. In order to prove that 01-Permanent is #P-hard, it is therefore sufficient to show that the number of satisfying assignments for a 3-CNF formula can be expressed succinctly as a function of the permanent of a matrix that contains only the values 0 and 1. This is usually accomplished in two steps: # Given a 3-CNF formula φ, construct a directed integer-weighted graph G_\phi, such that the sum of the weights of cycle covers of G_\phi (or equivalently, the permanent of its adjacency matrix) is equal to the number of satisfying assignments of φ. This establishes that Permanent is #P-hard.
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Murder site of and small memorial for Mahad Mohammed on the Winstanley Estate for The Landscape of Murder Photo ProjectThe modern estates were notorious within the local police station according to Mike Fuller, the first Black Chief Constable, Commander and Deputy Assistant Commissioner within the UK, who was Chief Superintendent at Battersea Police Station between 1999–2002. The police implemented official advice never to enter the estates alone because of the "widespread dealing" where dealers "had no fear of anyone, including the police". Policemen were also advised never to walk under walkways, where "anything could be dropped on your head" that also functioned as escape routes for criminals. The adjacency of the estates to Clapham Junction have made them an attractive location for drug dealing, which gained local notoriety because of the decision to introduce harsher penalties on the estate as a trial initiative in 2007.
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In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time as the determinant of the Laplacian matrix of the graph. It is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph. Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph that is equal to the difference between the graph's degree matrix (a diagonal matrix with vertex degrees on the diagonals) and its adjacency matrix (a (0,1)-matrix with 1's at places corresponding to entries where the vertices are adjacent and 0's otherwise). For a given connected graph G with n labeled vertices, let λ1, λ2, ..., λn−1 be the non- zero eigenvalues of its Laplacian matrix.
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Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17. A family of graphs has a universal graph of polynomial size, containing every -vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by -bit bitstrings such that an algorithm can determine whether two vertices are adjacent by examining their labels. For, if a universal graph of this type exists, the vertices of any graph in may be labeled by the identities of the corresponding vertices in the universal graph, and conversely if a labeling scheme exists then a universal graph may be constructed having a vertex for every possible label.. In older mathematical terminology, the phrase "universal graph" was sometimes used to denote a complete graph. The notion of universal graph has been adapted and used for solving mean payoff games.
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A sentence S may be true for some graphs, and false for others; a graph G is said to model S, written G\models S, if S is true of the vertices and adjacency relation of G. The algorithmic problem of model checking concerns testing whether a given graph models a given sentence. The algorithmic problem of satisfiability concerns testing whether there exists a graph that models a given sentence. Although both model checking and satisfiability are hard in general, several major algorithmic meta-theorems show that properties expressed in this way can be tested efficiently for important classes of graphs. Other topics of research in the logic of graphs include investigations of the probability that a random graph has a property specified within a particular type of logic, and methods for data compression based on finding logical formulae that are modeled by a unique graph.
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For example, the Baltimore Beltway in Maryland crosses Baltimore Harbor on a high arch bridge, and much of the partially completed Stockholm Ring Road in Sweden runs through tunnels or over long bridges. Some towns or cities on seacoasts or near rugged mountains cannot have a full ring road, such as Dublin's ring road; or, in the US, Interstate 287, mostly in New Jersey (bypassing New York City), or Interstate 495 around Boston, none of which completely circles these seaport cities. In other cases, adjacency of international boundaries may prevent ring road completion. Construction of a true ring road around Detroit is effectively blocked by its location on the border with Canada; although constructing a route mostly or entirely outside city limits is technically feasible, a true ring around Detroit would necessarily pass through Canada, and so Interstate 275 and Interstate 696 together bypass but do not encircle the city.
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One reason for interest in the computational complexity of the permanent is that it provides an example of a problem where constructing a single solution can be done efficiently but where counting all solutions is hard. As Papadimitriou writes in his book Computational Complexity: Specifically, computing the permanent (shown to be difficult by Valiant's results) is closely connected with finding a perfect matching in a bipartite graph, which is solvable in polynomial time by the Hopcroft–Karp algorithm.John E. Hopcroft, Richard M. Karp: An n^{5/2} Algorithm for Maximum Matchings in Bipartite Graphs. SIAM J. Comput. 2(4), 225–231 (1973) For a bipartite graph with 2n vertices partitioned into two parts with n vertices each, the number of perfect matchings equals the permanent of its biadjacency matrix and the square of the number of perfect matchings is equal to the permanent of its adjacency matrix.
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A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron) In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.. Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.
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The axon hillock and initial segment have a number of specialized properties that make them capable of action potential generation, including adjacency to the axon and a much higher density of voltage-gated ion channels than is found in the rest of the cell body. In dorsal root ganglion cells, the cell body is thought to have approximately 1 voltage-gated sodium channel per square micrometre, while the axon hillock and initial segment of the axon have about ~100–200 voltage-gated sodium channels per square micrometre; in comparison, the nodes of Ranvier along the axon are thought to have ~1000–2000 such channels per square micrometre. This clustering of voltage-gated ion channels is a consequence of plasma-membrane and cytoskeletal associating proteins such as ankyrin. In electrophysiological models, the axon hillock is included with the initial segment of the axon where membrane potentials propagated from synaptic inputs to the dendrites or cell body are summed.
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A custom-made ankle/foot orthosis for the treatment of patients having plantar ulcers is disclosed, which comprises a rigid L-shaped support member and a rigid anterior support shell hingedly articulated to the L-shaped support member. The plantar portion of the L-shaped member further comprises at least one ulcer-protecting hollow spatially located for fitted placement in inferior adjacency to a user's plantar ulcer, thus allowing the user to transfer the user's weight away from the plantar ulcer and facilitating plantar ulcer treatment. The anterior support shell is designed for lateral hinged attachment to the L-shaped member to take advantage of medial tibial flare structure for enhancing the weight- bearing properties of the disclosed orthosis. A flexible, polyethylene hinge member hingedly attaches the anterior support shell to the L-shaped member and securing straps securely attach the anterior support shell in fixed, weight- bearing relation about the proximal, anterior portion of the user's lower leg.
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Retraction of a cube onto a six-vertex subgraph. A retraction of a graph G is an adjacency-preserving map from G to one of its subgraphs.. More precisely, it is graph homomorphism φ from G to itself such that φ(v) = v for each vertex v in the subgraph φ(G). The image of the retraction is called a retract of G. Retractions are examples of metric maps: the distance between φ(v) and φ(w), for every v and w, is at most equal to the distance between v and w, and is equal whenever v and w both belong to φ(G). Therefore, a retract must be an isometric subgraph of G: distances in the retract equal those in G. If G is a median graph, and a, b, and c are an arbitrary three vertices of a retract φ(G), then φ(m(a,b,c)) must be a median of a, b, and c, and so must equal m(a,b,c).
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Robertson's example of a non-YΔY-reducible apex graph. A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a Δ-Y or Y-Δ transform, the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its two neighboring edges by a single edge.. Like the apex graphs and the linkless embeddable graphs, the YΔY-reducible graphs are closed under graph minors. And, like the linkless embeddable graphs, the YΔY-reducible graphs have the seven graphs in the Petersen family as forbidden minors, prompting the question of whether these are the only forbidden minors and whether the YΔY-reducible graphs are the same as the linkless embeddable graphs. However, Neil Robertson provided an example of an apex graph that is not YΔY-reducible.
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The notion of an implicit graph is common in various search algorithms which are described in terms of graphs. In this context, an implicit graph may be defined as a set of rules to define all neighbors for any specified vertex.. This type of implicit graph representation is analogous to an adjacency list, in that it provides easy access to the neighbors of each vertex. For instance, in searching for a solution to a puzzle such as Rubik's Cube, one may define an implicit graph in which each vertex represents one of the possible states of the cube, and each edge represents a move from one state to another. It is straightforward to generate the neighbors of any vertex by trying all possible moves in the puzzle and determining the states reached by each of these moves; however, an implicit representation is necessary, as the state space of Rubik's Cube is too large to allow an algorithm to list all of its states.
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The most common approach to resource management across languages is to use unwind protection, which is called when execution exits a scope – by execution running off the end of the block, returning from within the block, or an exception being thrown. This works for stack-managed resources, and is implemented in many languages, including C#, Common Lisp, Java, Python, Ruby, and Scheme. The main problems with this approach is that the release code (most commonly in a `finally` clause) may be very distant from the acquisition code (it lacks adjacency), and that the acquisition and release code must always be paired by the caller (it lacks encapsulation). These can be remedied either functionally, by using closures/callbacks/coroutines (Common Lisp, Ruby, Scheme), or by using an object that handles both the acquisition and release, and adding a language construct to call these methods when control enters and exits a scope (C# `using`, Java `try`-with-resources, Python `with`); see below.
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The wages paid to these workers is higher due to the undesirable nature of their professions. However, in regions where certain classes of workers are restricted to this type of work or there are contributing regional conditions - for example, high unemployment, adjacency to regions with high poverty, or those that are recipient of driven labor migration - there will be workers willing to accept lower than equilibrium wages and then these jobs are not well paid by any definition. Large scale international labor migration, from developing to developed countries since the late 19th century and early 20th century has provided a pool of migrants willing to undertake employment for lower wages than native residents. Higher wages in developed countries are a strong 'pull' factor in international migration, and thus while a migrant worker is willing to accept a comparatively low-wage for a 3D job in a developed country it may mean a significant increase in wages compared to their originating country.
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In computational complexity theory, SNP (from Strict NP) is a complexity class containing a limited subset of NP based on its logical characterization in terms of graph-theoretical properties. It forms the basis for the definition of the class MaxSNP of optimization problems. It is defined as the class of problems that are properties of relational structures (such as graphs) expressible by a second-order logic formula of the following form: : \exists S_1 \dots \exists S_\ell \, \forall v_1 \dots \forall v_m \,\phi(R_1,\dots,R_k,S_1,\dots,S_\ell,v_1,\dots,v_m), where R_1,\dots,R_k are relations of the structure (such as the adjacency relation, for a graph), S_1,\dots,S_\ell are unknown relations (sets of tuples of vertices), and \phi is a quantifier-free formula: any boolean combination of the relations. That is, only existential second-order quantification (over relations) is allowed and only universal first-order quantification (over vertices) is allowed.
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For a simple graph with vertex set , the adjacency matrix is a square matrix such that its element is one when there is an edge from vertex to vertex , and zero when there is no edge.. The diagonal elements of the matrix are all zero, since edges from a vertex to itself (loops) are not allowed in simple graphs. It is also sometimes useful in algebraic graph theory to replace the nonzero elements with algebraic variables.. The same concept can be extended to multigraphs and graphs with loops by storing the number of edges between each two vertices in the corresponding matrix element, and by allowing nonzero diagonal elements. Loops may be counted either once (as a single edge) or twice (as two vertex-edge incidences), as long as a consistent convention is followed. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention.
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The bipartite double cover of G has two vertices ui and wi for each vertex vi of G. Two vertices ui and wj are connected by an edge in the double cover if and only if vi and vj are connected by an edge in G. For instance, below is an illustration of a bipartite double cover of a non-bipartite graph G. In the illustration, each vertex in the tensor product is shown using a color from the first term of the product (G) and a shape from the second term of the product (K2); therefore, the vertices ui in the double cover are shown as circles while the vertices wi are shown as squares. :Image:Covering-graph-2.svg The bipartite double cover may also be constructed using adjacency matrices (as described below) or as the derived graph of a voltage graph in which each edge of G is labeled by the nonzero element of the two-element group.
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The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established by the facts that side BC corresponds to (in this case of congruence, equals in length) JK, CD corresponds to KL, DA corresponds to LI, and AB corresponds to IJ, while angle C corresponds to (equals) angle K, D corresponds to L, A corresponds to I, and B corresponds to J. In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has sequential sides a, b, c, d, and e and the other has sequential sides v, w, x, y, and z, and if b and w are corresponding sides, then side a (adjacent to b) must correspond to either v or x (both adjacent to w).
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It is not difficult to show that the conditions of the Erdős–Gallai theorem are necessary for a sequence of numbers to be graphic. The requirement that the sum of the degrees be even is the handshaking lemma, already used by Euler in his 1736 paper on the bridges of Königsberg. The inequality between the sum of the k largest degrees and the sum of the remaining degrees can be established by double counting: the left side gives the numbers of edge-vertex adjacencies among the k highest-degree vertices, each such adjacency must either be on an edge with one or two high- degree endpoints, the k(k-1) term on the right gives the maximum possible number of edge-vertex adjacencies in which both endpoints have high degree, and the remaining term on the right upper bounds the number of edges that have exactly one high degree endpoint. Thus, the more difficult part of the proof is to show that, for any sequence of numbers obeying these conditions, there exists a graph for which it is the degree sequence.
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If existential quantification over vertices were also allowed, the resulting complexity class would be equal to NP (more precisely, the class of those properties of relational structures that are in NP), a fact known as Fagin's theorem. For example, SNP contains 3-Coloring (the problem of determining whether a given graph is 3-colorable), because it can be expressed by the following formula: : \exists S_1 \exists S_2 \exists S_3 \, \forall u \forall v \, \bigl( S_1(u) \vee S_2(u) \vee S_3(u) \bigr) \, \wedge \, \bigl( E(u,v)\,\implies\,( eg S_1(u) \vee eg S_1(v))\,\wedge\,\left( eg S_2(u) \vee eg S_2(v)\right)\,\wedge\,( eg S_3(u) \vee eg S_3(v)) \bigr) . Here E denotes the adjacency relation of the input graph, while the sets (unary relations) S_1,S_2,S_3 correspond to sets of vertices colored with one of the 3 colors. Similarly, SNP contains the k-SAT problem: the boolean satisfiability problem (SAT) where the formula is restricted to conjunctive normal form and to at most k literals per clause, where k is fixed.
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A tree data structure can be defined recursively as a collection of nodes (starting at a root node), where each node is a data structure consisting of a value, together with a list of references to nodes (the "children"), with the constraints that no reference is duplicated, and none points to the root. Alternatively, a tree can be defined abstractly as a whole (globally) as an ordered tree, with a value assigned to each node. Both these perspectives are useful: while a tree can be analyzed mathematically as a whole, when actually represented as a data structure it is usually represented and worked with separately by node (rather than as a set of nodes and an adjacency list of edges between nodes, as one may represent a digraph, for instance). For example, looking at a tree as a whole, one can talk about "the parent node" of a given node, but in general as a data structure a given node only contains the list of its children, but does not contain a reference to its parent (if any).
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