In this chapter we collect the results on graphs we will need to study groups generated by 3-transpositions. One such result consists of the list of standard numerical constraints on strongly regular graphs, which appears as Theorem 6.3. In the context of the graph of a rank 3 permutation group, this was first proved by D. Higman in [Hi]. A proof of Theorem 6.3 is contained in Section 16 of [FGT], so no proof is included here. There is also a brief discussion of the lines defined by a graph.

Finally most effort is devoted to the notion of a contraction of a graph, particularly as applied to the commuting graph of a locally conjugate conjugacy class of a finite group. This material comes from [A2] and [AH]. It is an abstraction of ideas introduced by Fischer in Section 1 and 3 of [Fl] and [F2], most particularly Theorem 3.3.5 of those references.

Graphs

A graph is a pair Γ = (V, E) where V is a set of vertices (or points or objects) and E is a symmetric relation on V called adjacency (or incidence or something else). The ordered pairs in the relation E are called the edges of the graph. We say u is adjacent to v if (u, v) ∈ E is an edge in Γ.