Zeta functions of directed graphs

DEFINITION 12.1   A directed graph (digraph) is a pair $X^o=(V^o,E^o)$ of:

• a set $V^o$ , whose elements are called vertices or nodes,
• a set $E^o \subset V^o \times V^o$ of ordered pairs of vertices, called directed edges.

By definition, each element $e$ of $E^o$ is a pair $e=(s,t)$ of elements of $V^o$ . let us call $s$ (resp. $t$ ) the source (resp. the target) of $e$ .

Let $X^o=(V^o,E^o)$ be a directed graph. For each positive integer $m$ , we let $N_m$ to be the number of admissible closed paths in $X^o$ . Then we put

$$Z^o_{X^o} (T)= \exp\left( \sum \frac{1}{m} N_m T^m \right)$$

We define Perron Frobeinus operator $L_{X^o}:C(V^o)\to C(V^o)$ to be

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

PROPOSITION 12.2  

$$Z^o_{X^o}(T) = \frac{1}{\det(1-T\cdot L_{X^o})}$$

EXERCISE 12.1   Let $(M,\varphi)$ be a dinamical system of a finite set $M$ . Is it possible to define a directed graph $X=(M,E)$ such that its zeta function $Z_X$ coincides with the zeta function of the dinamical system $(M,\varphi)$ ? (Compare the Perron Frobenius matrix $L_{X}$ with `the pull back matrix' $P_\varphi$ .)