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Table Of Vertex-symmetric Digraphs
The best known vertex transitive digraphs (as of October 2008) in the directed Degree diameter problem In graph theory, the degree diameter problem is the problem of finding the largest possible graph (in terms of the size of its vertex set ) of diameter such that the largest degree of any of the vertices in is at most . The size of is bounde ... are tabulated below. Table of the orders of the largest known vertex-symmetric graphs for the directed degree diameter problem Key to colors References * * * * * * {{citation , last1 = Loz , first1 = Eyal , last2 = Širáň , first2 = Jozef , title = New record graphs in the degree-diameter problem , journal = Australasian Journal of Combinatorics , volume = 41 , year = 2008 , pages = 63–80 , url = http://ajc.maths.uq.edu.au/pdf/41/ajc_v41_p063.pdf External links Vertex-symmetric Digraphsonline table. The Degree - Diameter Problem on CombinatoricsWiki.org Eyal Loz'sDegree-Diamet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex-transitive Graph
In the mathematical field of graph theory, a vertex-transitive graph is a graph in which, given any two vertices and of , there is some automorphism :f : G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices.. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph). Finite examples Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Diameter Problem
In graph theory, the degree diameter problem is the problem of finding the largest possible graph (in terms of the size of its vertex set ) of diameter such that the largest degree of any of the vertices in is at most . The size of is bounded above by the Moore bound; for and only the Petersen graph, the Hoffman-Singleton graph, and possibly one more graph (not yet proven to exist) of diameter and degree attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound. Formula Let n_ be the maximum possible number of vertices for a graph with degree at most ''d'' and diameter ''k''. Then n_\leq M_, where M_ is the Moore bound: :M_=\begin1+d\frac&\textd>2\\2k+1&\textd=2\end This bound is attained for very few graphs, thus the study moves to how close there exist graphs to the Moore bound. For asymptotic behaviour note that M_=d^k+O(d^). Define the parameter \mu_k=\liminf_\frac. It is conjectured that \mu_k=1 for all ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
