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...

Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle ~ C of Γ is called G-consistent whenever there is an element of G whose restriction to ~ C is the 1-step rotation of ~ C. Consistent cycles in finite arc-transitive graphs were introduced by Conway in one of his public lectures. He observed that the number of G-orbits of G-consistent cycles of an ...

A (di)graph is supereulerian if it contains a spanning, connected, eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we give a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the...

This paper develops methods to reason about graph transformations, and in particular to show that transitivity and reachability invariants are preserved during transformations. In our approach, graph transformations consist of a pattern defining an applicability condition, and an operational description of the desired transformation. Whereas previous work was restricted to Boolean combinations ...

The notion of a C-ultrahomogeneous (or C-UH) graph due to D. Isaksen et al. is adapted for digraphs and applied to the cubic distance-transitive graphs considered both as graphs and digraphs when C is formed by shortest cycles and (k − 1)-paths with k = arc-transitivity. Moreover, (k − 1)powers of shortest cycles taken with orientation assignments that make these graphs become C-UH digraphs are...

Let A be the adjacency matrix of a graph Γ . The nullity of A (that is, the dimension of the nullspace of A), when viewed as a matrix over a field of prime characteristic p, is called the p-nullity of Γ . We present several families of arc-transitive graphs with arbitrarily large p-nullity. We also show that the p-nullity of a vertextransitive graph of order a power of p is zero, provided that ...

A homogeneous factorisation of a digraph Γ consists of a partition P = {P1, . . . , Pk} of the arc set AΓ and two vertex-transitive subgroups M 6 G 6 Aut(Γ) such that M fixes each Pi setwise while G leaves P invariant and permutes its parts transitively. Given two graphs Γ1 and Γ2 we consider several ways of taking a product of Γ1 and Γ2 to form a larger graph, namely the direct product, cartes...

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs.

The transitive closure (or reachability) problem in a directed graph consists in finding whether there is a path between any two vertices. In this paper, we first study the problem of parallelization of Italiano's algorithm for dynamic updating the transitive closure after inserting a new arc into the graph represented as a list of arcs. To this end, by means of the data structure first propose...