The Johnson graph J(v, k) has, as vertices, the k-subsets of a v-set V and as edges the pairs of k-subsets with intersection of size k − 1. We introduce the notion of a neighbour-transitive code in J(v, k). This is a vertex subset Γ such that the subgroup G of graph automorphisms leaving Γ invariant is transitive on both the set Γ of ‘codewords’ and also the set of ‘neighbours’ of Γ, which are ...

For a positive integer n, does there exist a vertex-transitive graph r on n vertices which is not a Cayley graph, or, equivalently, a graph r on n vertices such that Aut F is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, MaruSic and Scapellato, and McKay and the second author) has produced answers to ...

We study locally s-arc-transitive graphs arising from the quasiprimitive product action (PA). prove that, for any (G,2)-arc-transitive graph with G acting quasiprimitively type PA on both G-orbits of vertices, group does not act primitively either orbit. Moreover, we construct first examples that are standard double covers type, answering existence question these graphs.

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. In this chapter we explore some of the ways the two theories have influenced each other. On the one hand each finite transitive permutation group corresponds to several vertex-transitive graphs, namely the generalised orbital graphs which we shall discuss below. On the other hand, ...

In this sequel to the paper ‘Arc-transitive abelian regular covers of cubic graphs’, all arc-transitive abelian regular covers of the Heawood graph are found. These covers include graphs that are 1-arc-regular, and others that are 4-arc-regular (like the Heawood graph). Remarkably, also some of these covers are 2-arc-regular.

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...

Let r be a finite connecied regular bipartite 2-arc transitive graph. It is shown that r is a cover of a possibly smaller graph :E, which is also connecied and regular of the same valency as r, and there is a subgroup G of Aut :E such that G is 2-arc transitive on :E and every nontrivial normal subgroup of G has at most two orbits on vertices. Such graphs :E for which the subgroup G has an abel...

‎A set of vertices $S$ of a graph $G$ is called a fixing set of $G$‎, ‎if only the trivial automorphism of $G$ fixes every vertex in $S$‎. ‎The fixing number of a graph is the smallest cardinality of a fixing‎ ‎set‎. ‎The fixed number of a graph $G$ is the minimum $k$‎, ‎such that ‎every $k$-set of vertices of $G$ is a fixing set of $G$‎. ‎A graph $G$‎ ‎is called a $k$-fixed graph‎, ‎if its fix...

A transitive orientation of an undirected graph is an assignment of directions to itsedges so that these directed edges represent a transitive relation between the vertices ofthe graph. Not every graph has a transitive orientation, but every graph can be turnedinto a graph that has a transitive orientation, by adding edges. We study the problem ofadding an inclusion minimal set ...