Primes and ihara zeta function on graphs

Question

The ihara zeta function of a graph $X$ is defined as $$\zeta_X(u)=\prod_{ [C] }(1-u^{v(C)})$$ where the product is over the primes of the graph( A.Terras Zeta functions of graphs a stroll through the Garden)

The question is : Can we take two graphs with the same ihara but different multiset of lengths of primes? Maybe the answer is obvious but i can't see it. If yes i would like to see an example.

Answer 1

No, this can't happen. You would have $$ 1=\prod_{n}(1-u^n)^{k_n}, $$ where $k_n\in\mathbb Z$ is the difference of the number of primes in the first graph of length $n$ minus the number in the second graph. Applying the logarithm you get $ 0=\sum_{m}c_m u^m, $ where $$ c_m=-\sum_{d|m}dk_d/m. $$ Which implies $k_n=0$ for all $n$.