Next we want to consider the Ihara zeta function for regular graphs (which are unweighted and which satisfy our usual hypotheses, for the most part). We need some facts from graph theory first. References for the subject include Biggs [15], Bollobas [18], [19], Cvetković, Doob, and Sachs [32].

Definition 7.1 A graph is a bipartite graph iff its set of vertices can be partitioned into two disjoint sets S, T such that no vertex in S is adjacent to any other vertex in S and no vertex in T is adjacent to any other vertex in T.

Exercise 7.1 Show that the cube of Figure 2.8 is an example of a bipartite graph.

Proposition 7.2 (Facts about Spectrum A, when A is the adjacency operator of a connected (q + 1)-regular graph X) Assume that X is a connected (q + 1)-regular graph and that A is its adjacency matrix. Then:

1. (1) λ ∈ Spectrum A implies that ∣λ∣ ≤ q + 1;

2. (2) q + 1 ∈ Spectrum A and has multiplicity 1;

3. (3) −(q + 1) ∈ Spectrum A iff the graph X is bipartite.

Proof of fact (1) Note that q + 1 is clearly an eigenvalue of A corresponding to the constant vector. Suppose that Aυ = λυ, for some column vector υ = t(υ1 … υn) ∈ ℝn, and suppose that the maximum ∣υi∣ occurs at i = a.