“Almost divisibility” in the Ihara zeta functions of certain ramified covers of [formula omitted]-regular graphs

Abstract

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.