The usual hypotheses and some definitions

Our graphs will be finite, connected, and undirected. It will usually be assumed that they contain no degree-1 vertices, called “leaves” or “hair” or “danglers”. We will also usually assume that the graphs are not cycles or cycles with hair. A cycle graph is obtained by arranging the vertices in a circle and connecting each vertex to the two vertices next to it on the circle. A “bad” graph – meaning that it does not satisfy the above assumptions – is pictured in Figure 2.1. We will allow our graphs to have loops and multiple edges between pairs of vertices.

Why do we make these assumptions? They are necessary hypotheses for many of the main theorems (for example, the graph theory prime number theorem, formula (2.4)). References for graph theory include Biggs [15], Bollobás [19], Fan Chung [26], and Cvetković, Doob, and Sachs [32].

A regular graph is a graph each of whose vertices has the same degree, i.e., the same number of edges coming out of the vertex. A graph is k-regular if every vertex has degree k. Simple graphs have no loops or multiple edges. Our graphs need not be regular or simple. A complete graphKn on n vertices has all possible edges between its vertices but no loops.

Definition 2.1 Let V denote the vertex set of a graph X with n =∣V∣. The adjacency matrixA of X is an n × n matrix with (i, j) th entry