Ihara zeta function

Let $X$ be a finite graph. The Ihara zeta function is given by the formula $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the Euler product representation of the Riemann zeta function.

References

The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps - Rostislav I. Grigorchuk, Andrzej Zuk What are zeta functions of graphs and what are they good for? - Matthew D. Horton, H.M. Stark, Audrey A. Terras