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?

Sorry I should have elaborated more.

I know of the formula $$ c_{2m}=(-1)^{m-n}\prod_{v_i \in V}(\deg(v_i)-1)$$ given in a PhD thesis. I am looking for more down to earth cycle interpretation of $c_{2m}$ and $c_{i}$ for $i>|E|$ in general.

I am aware of the document http://msp.org/involve/2008/1-2/involve-v1-n2-p08-p.pdf I just do not understand the interpretation of $c_{2m}$ for cycle graphs there since no general interpretation is given for $c_{i}$ for $i>|E|$ in general. Only interpretation seems to be twice sum of disjoint cycles or four times sum of overlapping cycles whose sum is $2m$. However twice or four times an integral value are greater than $1$ which is the value of $c_{2m}$ for cycle graphs.