A subgroup of the automorphism group a graph Γ is said to be half-arc-transitive on if its action transitive vertex set and edge but not arc Γ. Tetravalent graphs girths 3 4 admitting automorphisms have previously been characterized. In this paper we study examples girth 5. We show that, with two exceptions, all such only directed 5-cycles respect corresponding induced orientation edges. Moreov...

Let G be a finite group, and X a subset of G. The Cayley graph of G with respect to X , written Cay(G,X) has two different definitions in the literature. The vertex set of this graph is the group G. In one definition, there is an arc from g to xg for all g ∈ G and x ∈ X ; in the other definition, for the same pairs (g,x), there is an arc from g to gx. Cayley graphs are generalised by coset grap...

A transitive decomposition is a pair ðG;PÞ where G is a graph and P is a partition of the arc set of G, such that there exists a group of automorphisms of G which leaves P invariant and transitively permutes the parts in P. This paper concerns transitive decompositions where the group is a primitive rank 3 group of ‘grid’ type. The graphs G in this case are either products or Cartesian products...

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...