All Ihara $\zeta$ functions for planar $k$-regular graphs with a given set of faces are equivalent
Solution 1:
Like Chris Godsil says in the comments, for regular graphs, the Ihara zeta function contains the same information as the (multi)set of eigenvalues of the adjacency matrix. So two $k$-regular graphs have the same Ihara zeta function iff they're isospectral.