For a connected graph G = V , E , let id="M2"> v ∈ be vertex and id="M3"> e uw an edge. The distance between the id="M4"> edge id="M5"> is given by id="M6"> d</mi...

are considered in this paper. Examples of finite Qn-geometries are constructed for any n i> 2. In particular, we obtain new examples of finite 2-arc-transitive graphs of girth 5, containing Petersen subgraphs (cf. [4], [5] for an enquiry into this class of graphs). Furthermore, we give a complete classification of Q2-geometries using coset enumeration. In particular, we show that they are all f...

These are notes from lectures given in the Queen Mary Combinatorics Study 1 Introductory Definitions In these notes, Γ = (V Γ, EΓ) denotes a graph, as we will use G to denote a group. All graphs considered will be simple, finite, connected and undirected. Definition An automorphism of Γ is a bijective function g : V Γ → V Γ such that v ∼ w if and only if g(v) ∼ g(w). The set of all automorphism...

A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159–162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D ∪ T necessarily contains all vertices of the graph.