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Forcing Graphs
In graph theory, a forcing graph is one whose density determines whether a graph sequence is quasi-random. The term was first coined by Chung, Graham, and Wilson in 1989., and forcing graphs play an important role in the study of pseudorandomness in graph sequences. Definitions Let , known as the subgraph density (in particular, is the edge density of ). A sequence of graphs is called quasi-random if, for all graphs , the edge density approaches some and approaches as increases, where is the number of edges in . Intuitively, this means that a graph sequence with a given edge density has the number of graph homomorphisms that one would expect in a random graph sequence. A graph is called forcing if for all graph sequences where approaches as goes to infinity, is quasi-random if approaches . In other words, one can verify that a sequence of graphs is quasi-random by just checking the homomorphism density of a single graph. There is a second definition of forcing g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by ''edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Graph
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete definition of graph pseudorandomness, but there are many reasonable characterizations of pseudorandomness one can consider. Pseudorandom properties were first formally considered by Andrew Thomason in 1987. He defined a condition called "jumbledness": a graph G=(V,E) is said to be (p,\alpha)-''jumbled'' for real p and \alpha with 0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism Density
In the mathematical field of extremal graph theory, homomorphism density with respect to a graph H is a parameter t(H,-) that is associated to each graph G in the following manner: : t(H,G):=\frac. Above, \operatorname(H,G) is the set of graph homomorphisms, or adjacency preserving maps, from H to G. Density can also be interpreted as the probability that a map from the vertices of H to the vertices of G chosen uniformly at random is a graph homomorphism. There is a connection between homomorphism densities and subgraph densities, which is elaborated on below. Examples * The edge density of a graph G is given by t(K_,G). * The number of walks with k-1 steps is given by \operatorname(P_k, G). *\operatorname(C_k, G) = \operatorname(A^k) where A is the adjacency matrix of G. *The proportion of colorings using k colors that are proper is given by t(G, K_k). Other important properties such as the number of stable sets or the maximum cut can be expressed or estimated in terms of h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphon
GraphOn GO-Global is a multi-user remote access application for Windows. Overview GO-Global allows multiple users to concurrently run Microsoft Windows applications installed on a Windows server or server farm from network-connected locations and devices. GO-Global redirects the user interface of Windows applications running on the Windows server to the display or browser on the user's device. Applications look and feel like they are running on the user's device. Supported end-user devices include Windows, Mac and Linux personal computers, iOS and Android mobile devices, and Chromebooks. GO-Global is used by Independent Software Vendors (ISVs), Hosted Service Providers (HSPs), and Managed Service Providers (MSPs) to publish Windows applications without modification of existing code for the use of local and remote users. Architecture GO-Global enables multi-user remote access to Windows applications without the use of Microsoft Remote Desktop Services (RDS) or th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Rényi Model
In the mathematical field of graph theory, the Erdős–Rényi model is either of two closely related models for generating random graphs or the evolution of a random network. They are named after Hungarian mathematicians Paul Erdős and Alfréd Rényi, who first introduced one of the models in 1959, while Edgar Gilbert introduced the other model contemporaneously and independently of Erdős and Rényi. In the model of Erdős and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely; in the model introduced by Gilbert, also called the Erdős–Rényi–Gilbert model, each edge has a fixed probability of being present or absent, independently of the other edges. These models can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs. Definition There are two closely related variants of the Erdős ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sidorenko's Conjecture
Sidorenko's conjecture is a conjecture in the field of graph theory, posed by Alexander Sidorenko in 1986. Roughly speaking, the conjecture states that for any bipartite graph H and graph G on n vertices with average degree pn, there are at least p^ n^ labeled copies of H in G, up to a small error term. Formally, it provides an intuitive inequality about graph homomorphism densities in graphons. The conjectured inequality can be interpreted as a statement that the density of copies of H in a graph is asymptotically minimized by a random graph, as one would expect a p^ fraction of possible subgraphs to be a copy of H if each edge exists with probability p. Statement Let H be a graph. Then H is said to have Sidorenko's property if, for all graphons W, the inequality : t(H,W)\geq t(K_2,W)^ is true, where t(H,W) is the homomorphism density of H in W. Sidorenko's conjecture (1986) states that every bipartite graph has Sidorenko's property. If W is a graph G, this means that the pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
