|
Henson Graph
In graph theory, the Henson graph is an undirected infinite graph, the unique countable homogeneous graph that does not contain an -vertex clique but that does contain all -free finite graphs as induced subgraphs. For instance, is a triangle-free graph that contains all finite triangle-free graphs. These graphs are named after C. Ward Henson, who published a construction for them (for all ) in 1971.. The first of these graphs, , is also called the homogeneous triangle-free graph or the universal triangle-free graph. Construction To construct these graphs, Henson orders the vertices of the Rado graph into a sequence with the property that, for every finite set of vertices, there are infinitely many vertices having as their set of earlier neighbors. (Only the Rado graph has such a sequence.) He then defines to be the induced subgraph of the Rado graph formed by removing the final vertex (in the sequence ordering) of every -clique of the Rado graph. With this construction, eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Model Property
In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some ''finite'' model of L. Another way of putting this is to say that L has the fmp if for every formula ''A'' of L, ''A'' is an L-theorem if and only if ''A'' is a theorem of the theory of finite models of L. If L is finitely axiomatizable (and has a recursive set of inference rules) and has the fmp, then it is decidable. However, the result does not hold if L is merely recursively axiomatizable. Even if there are only finitely many finite models to choose from (up to isomorphism) there is still the problem of checking whether the underlying frames of such models validate the logic, and this may not be decidable when the logic is not finitely axiomatizable, even when it is recursively axiomatizable. (Note that a logic is recursively enumerable if and only if it is recursively axiomatizable, a result known as Craig's theorem.) Example A first-order formula wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Graph
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I J K L M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Graph
In mathematics, a ''k''-ultrahomogeneous graph is a graph in which every isomorphism between two of its induced subgraphs of at most ''k'' vertices can be extended to an automorphism of the whole graph. A ''k''-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism). A homogeneous graph is a graph that is ''k''-homogeneous for every ''k'', or equivalently ''k''-ultrahomogeneous for every ''k'', and thus, every homogeneous graph is also ultrahomogeneous. It is a special case of a homogenous model. Classification The only finite homogeneous graphs are the cluster graphs ''mK''''n'' formed from the disjoint unions of isomorphic complete graphs, the Turán graphs formed as the complement graphs of ''mK''''n'', the 3 × 3 rook's graph, and the 5- cycle. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free Graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding or triangle detection 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 \tilde O\bigl(m^\bigr) where the \tilde O hides sub-polynomial factors. Here \omega is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific Journal Of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley. It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription. The journal is incorporated as a 501(c)(3) organization A 501(c)(3) organization is a United States corporation, Trust (business), trust, unincorporated association or other type of organization exempt from federal income tax under section 501(c)(3) of Title 26 of the United States Code. It is one of .... The 255-page proof of the odd order theorem, by Walter Feit and John Griggs Thompson, was publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rado Graph
In the mathematics, mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a Countable set, countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other. Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges, from the original graph, connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subseteq V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path between any two vertices in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Graph
In mathematics, a universal graph is an infinite Graph (discrete mathematics), graph that contains ''every'' finite (or at-most-countable) graph as an induced Glossary of graph theory#Subgraphs, subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph family : that is, an infinite graph belonging to that contains all finite graphs in . For instance, the Henson graphs are universal in this sense for the -clique-free graphs. A universal graph for a family of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in ; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph so a hypercube can be said to be a universal graph for trees. However it is not the smallest such graph: it is known that there is a universal graph for -vertex trees, with only vertices and edges, and that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
