Neighborhood Regular Graphs
نویسندگان
چکیده
A regular graph G is called vertex transitive if the automorphism group of G contains a single orbit. In this paper we define and consider another class of regular graphs called neighborhood regular graphs abbreviated NR. In particular, let G be a graph and N [v] be the closed neighborhood of a vertex v of G. Denote by G(N [v]) the subgraph of G induced by N [v]. We call G NR if G(N [v]) ∼= G(N [v′]) for each pair of vertices v and v′ in V (G). A vertex transitive graph is necessarily NR. The converse, however, is in general not true as is shown by the union of the cycles C4∪C5. Here we provide a method for constructing an infinite class of connected NR graphs which are not vertex transitive. A NR graph G is called neighborhood regular relative to N if N [v] ∼= N for each v ∈ V (G). Necessary conditions for N are given along with several theorems which address the problem of finding the smallest order (size) graph that is NR relative to a given N . A table of solutions to this problem is given for all graphs N up to order five.
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