Abstract We prove that, if $$\varGamma $$ Γ is a finite connected 3-valent vertex-transitive, or 4-valent vertex- and edge-transitive graph, then either part of well-understood family graphs, every non-identity automorphism fixes at most 1/3 the edges. This answers question proposed by Primož Potočnik third a...

Tian and Meng in [Y. Tian and J. Meng, c  -Optimally half vertex transitive graphs with regularity , Information Processing Letters 109 (2009) 683-686] shown that a connected half vertex transitive graph with regularity and girth is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree k k   6 g G   ...

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.

1. Introduction. A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group T in the family all graphs G whose vertex-set is the orbit V of T and for which T ^ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of c...

In this paper, a classification is given for tetravalent graphs of square-free order which are vertex-transitive and edge-transitive. It is shown that such graphs are Cayley graphs, edge-regular metacirculants and covers of some graphs arisen from simple groups A7, J1 and PSL(2, p).

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pair of parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs for certain families with prime power girth,...

Let p be a prime. It was shown by Folkman (J. Combin. Theory 3 (1967) 215) that a regular edge-transitive graph of order 2p or 2p is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that, with the exception of the Gray graph on 54 vertices, every cubic edge-transitive graph of order 2p is vertex-transitive. c © 2003 Elsev...

We give a necessary and su cient condition for a graphical regu lar representation to be adjacency transitive and provide an in nite family of nite simple undirected vertex transitive graphs such that neither nor c is adjacency transitive