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.



Primes and ihara zeta function on graphs
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 $n$.


