In this paper, we extend the notion of a circulant to a broader class of vertex.transitive graphs, which we call multidimensional circulants. This new class of graphs is shown to consist precisely of those vertextransitive graphs with an automorphism group containing a regular abelian subgroup. The result is proved using a theorem of Sabidussi which shows how to recover any vertex-transitive gr...

In this paper, we determine all of connected normal edge-transitive Cayley graphs on non-abelian groups with order $4p^2$, where $p$ is a prime number.

A subgroup G of automorphisms of a graph X is said to be 1 2-transitive if it is vertex and edge but not arc-transitive. The graph X is said to be 1 2-transitive if Aut X is 1 2-transitive. The graph X is called one-regular if Aut X acts regularly on the set arcs of X. The interplay of three diierent concepts of maps, one-regular graphs and 1 2-transitive group actions on graphs of valency 4 is...

Graphs possessing a high degree of symmetry have often been considered in topological graph theory. For instance, a number of constructions of genus embeddings by means of current or voltage graphs is based on the observation that a graph can be represented as a Cayley graph for some group. Another kind of embedding problems where symmetrical graphs are encountered is connected with regular map...

Several authors have studied methods to construct the transitive reduction of a directed graph, but little work has been done on how to maintain it. We are motivated by a real-world application which uses a transitively reduced graph at its core and must maintain the transitive reduction over a sequence of graph operations. This paper presents an efficient method to maintain the transitive redu...

The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on t...

The connection between doubly transitive permutation groups G on a finite set Cl which are not doubly primitive and automorphism groups of block designs in which X = 1 has been investigated by Sims [2] and Atkinson [1]. If, for a e Q, Ga has a set of imprimitivity of size 2 then it is easy to show that G is either sharply doubly transitive or is a group of automorphisms of a non-trivial block d...

The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on t...

We show that the critical value for the contact process on a vertex-transitive graph G with finitely many edges added to it is the same as the critical value for the contact process on G. This gives a partial answer to a conjecture of Pemantle and Stacey.

We study two transitivity properties for group actions on buildings, called Weyl transitivity and strong transitivity. Following hints by Tits, we give examples involving anisotropic algebraic groups to show that strong transitivity is strictly stronger than Weyl transitivity. A surprising feature of the examples is that strong transitivity holds more often than expected.