“Almost divisibility” in the Ihara zeta functions of certain ramified covers of q+1-regular graphs

Given a (q+1)-regular graph X and a second graph Y formed by taking k copies of X and identifying them at a common vertex, we form a ramified cover of the original graph. We prove that the reciprocal of the zeta function for X “almost divides” the reciprocal of the zeta function for Y, in the following sense. The reciprocal of the zeta function of X divides the product of the reciprocal of the zeta function of Y and some polynomial of bounded degree (which depends only on the graph X, not on the number of copies). Two specific examples show that in fact “almost divisibility” is the best that can be hoped for.