Ihara zeta function

DEFINITION 12.3   A graph (undirected graph) is a pair $X=(V,E)$ of:

• a set $V$ , whose elements are called vertices or nodes,
• a set $E \subset (V \times V)/\mathfrak{S}_2$ of unordered pairs of vertices, called edges.

Let $X=(V,E)$ be a graph. The Ihara zeta function of $X$ is defined by

$$Z(u)=\prod_{\mathfrak{p}\in P} (1-u^{\operatorname{length}(\mathfrak{p})})^{-1},$$

where $P$ denotes the set of prime cycles in $X$

We define the adjacency operator $A$ as

$$A(f)(x)=\sum_{\substack{e \in E\\ \operatorname{source}(e)=x}} f(\operatorname{target}(e)) =\sum_{(x,y)\in E} f(y) .$$

We also define the `degree operator' $D$ as:

$$D(f)(x)=\deg(x) f(x)$$

where the degree $\deg(x)$ of $x \in V$ is defined as

$$\deg(x)=\char93 \{e \in E; \operatorname{source}(e)=x\}.$$

THEOREM 12.4  

$$Z(u)= (1-u^2)^{\chi(X)} \det(I-u A + u^2(D-I))^{-1}$$

where $\chi(X)$ is the euler number of $X$ .