This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying such where automorphism group is an almost simple with socle alternating or sporadic group, and classical a vertex-stabiliser acting faithfully on set neighbours.

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

We study the class of edge-transitive graphs of square-free order and valency at most k. It is shown that, except for a few special families of graphs, only finitely many members in this class are basic (namely, not a normal multicover of another member). Using this result, we determine the automorphism groups of locally primitive arc-transitive graphs with square-free order.

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...

A graph is semi-symmetric if it is regular and edge transitive but not vertex transitive. The 3and 4-valent semi-symmetric graphs are wellstudied. Several papers describe infinite families of such graphs and their properties. 3-valent semi-symmetric graphs have been completely classified up to 768 vertices. The goal of this project is to extend this work to 5-valent semi-symmetric graphs. In th...

IS rel)re:se!otlng vertex-transitive as coset to allow us to obtain new results eral older results concerning the numbers transitive that are not 1 studied for more than a now, vertex-transitive graphs have quite great deal of activity. Their rich groups of automorphisms make from both the point of view of groups as well as of combinatorics. Among the most recent achievements, we can mention th...

A subgroup of the automorphism group a graph acts half-arc-transitively on if it transitively vertex-set and edge-set but not arc-set graph. If full half-arc-transitively, is said to be half-arc-transitive. In 1994 Gardiner Praeger introduced two families tetravalent arc-transitive graphs, called C±1 C±ε that play prominent role in characterization graphs admitting an automorphisms with normal ...

In this paper, we give a characterisation for class of edge-transitive Cayley graphs and provide method constructing valency 4 with arbitrarily large vertex stabiliser. particular, in the last section, obtain certain extensions results Li et al. (Tetravalent odd number vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.