Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime inor out-degree. Two differe...

A graph X is said to be 1 2-transitive if its automorphism group Aut X acts vertex-and edge-but not arc-transitively on X. Then Aut X induces an orientation of the edges of X. If X has valency 4, then this orientation gives rise to so called alternating cycles, that is even length cycles in X whose every other vertex is the head and every other vertex is the tail of its two incident edges in th...

We show that a complete arc K in the projective plane PG(2, q) admitting a transitive primitive group of projective transformations is either a cyclic arc of prime order or a known arc. If the completeness assumption is dropped, then K has either an affine primitive group, or K is contained in an explicit list. In order to find these primitive arcs, it is necessary to determine all complete k-a...

Quite a lot of attention has been paid recently to the construction of edgeor arc-transitive covers of symmetric graphs. In most cases, the approach has involved voltage graph techniques, which are excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian, or more generally, homocyclic, but are not so easy to use when the covering...

A graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. In this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

Abstract A connected graph of order n admitting a semiregular automorphism / k is called -multicirculant. Highly symmetric multicirculants small valency have been extensively studied, and several classification results exist for cubic vertex- arc-transitive multicirculants. In this paper, we study the broader class vertex-transitive graphs an /3 or larger that may not be semiregular. particular...

a graph $gamma$ is said to be vertex-transitive or edge‎- ‎transitive‎ ‎if the automorphism group of $gamma$ acts transitively on $v(gamma)$ or $e(gamma)$‎, ‎respectively‎. ‎let $gamma=cay(g,s)$ be a cayley graph on $g$ relative to $s$‎. ‎then, $gamma$ is said to be normal edge-transitive‎, ‎if $n_{aut(gamma)}(g)$ acts transitively on edges‎. ‎in this paper‎, ‎the eigenvalues of normal edge-tra...

In a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraph...

A cubic (trivalent) graph F is said to be 4-arc-transitive if its automorphism group acts transitively on the 4-arcs of r (where a 4-arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt-1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4(a), for m = 3 ,4 ,5 . . . , each presented in terms...

The generalised Paley graphs, as the name implies, are a generalisation of the Paley graphs which are well-known to be self-complementary and arc-transitive. They are defined as follows. Let F be a finite field of order p, where p is a prime, and let S be a subgroup of the multiplicative group of F, where S must have even order if p is odd. The generalised Paley graph of F relative to S has ver...