Assume we have a graph $G=(V,E)$.

The ihara zeta function $Z(G,u)$ is of form $$\frac1{\displaystyle\sum_{i=0}^{2|E|}c_iu^i}$$

A graph which has $|E|$ edges cannot have a simple cycle of length bigger than $|E|$.

So what do the coefficients $c_i$ mean for $i>|E|$?

In particular as an example what does $c_{2n}=1$ mean for an $n$-cycle graph?