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, ...

We associate a graph ΓG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x, y〉 is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the gro...

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Füredi and Komjáth, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.

We define a graph structure associated in natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields.

A $P_4$-free graph is called a cograph. In this paper we partially characterize finite groups whose power As will see, problem generalization of the determination in which every element has prime order, first raised by Graham Higman 1957 and fully solved very recently. First determine all $G$ $H$ for $G\times H$ We show that cograph can be characterised condition only involving elements orders ...