A classi cation of all arc transitive graphs of order a product of two primes is given Furthermore it is shown that cycles and complete graphs are the only arc transitive Cayley graph of abelian group of odd order Introductory remarks Throughout this paper graphs are nite simple and undirected By p and q we shall always denote prime numbers A k arc in a graph X is a sequence of k vertices v v v...

We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is generalization of the obtained by G. Chinta, J. Jorgenson and A. Karlsson from vertex-transitive graphs.

A graph is said to be half-arc-transitive if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called alternating cycles is associated, all of which have the same even length. Half of this length is called the radius of the graph in question. Moreover, any two adja...

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on graph only through degree and vertex-Cheeger constant. We also prove analogous result a large class irregular graphs, obtained as spanning subgraphs graphs. Using Babai, obtain operator in terms diameter degree.