A decomposition of a graph is a partition of the edge set. One can also look at partitions of the arc set but in this talk we restrict our attention to edges. If each part of the decomposition is a spanning subgraph then we call the decomposition a factorisation and the parts are called factors. Decompositions are especially interesting when the subgraphs induced by each part are pairwise isomo...

In this paper, we examine the structure of vertexand edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parame...

A regular and edge-transitive graph which is not vertex-transitive is said to be semisymmetric. Every semisymmetric graph is necessarily bipartite, with the two parts having equal size and the automorphism group acting transitively on each of these parts. A semisymmetric graph is called biprimitive if its automorphism group acts primitively on each part. In this paper biprimitive graphs of smal...

Abstract A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ transitive on the set of vertices and s -arcs , where for an integer $s \ge 1$ -arc a sequence $s+1$ $(v_0,v_1,\ldots ,v_s)$ such that $v_{i-1}$ $v_i$ are adjacent $1 i s$ $v_{i-1}\ne v_{i+1}$ s-1$ . 2-transitive it $(\text{Aut} ), 2)$ but not 3)$ -arc-transitive. Cayley group G normal in $\text{Aut} n...

It is proved that the Wiener index of a weighted graph (G, w) can be expressed as the sum of the Wiener indices of weighted quotient graphs with respect to an arbitrary combination of Θ∗-classes. Here Θ∗ denotes the transitive closure of the Djoković-Winkler’s relation Θ. A related result for edge-weighted graphs is also given and a class of graphs studied in [19] is characterized as partial cu...

We introduce a new class of graphs which we call P-transitive graphs, lying between transitive and 3-transitive graphs. First we show that the analogue of de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we show that the μ-calculus fixpoint hierarchy is infinite for Ptransitive graphs. Both results contrast with the case of transitive graphs. We give also an undecidabi...