The semicircle law for semiregular bipartite graphs

We give the (Ahumada type) Selberg trace formula for a semiregular bipartite graph G. Furthermore, we discuss the distribution on arguments of poles of zeta functions of semiregular bipartite graphs. As an application, we present two analogs of the semicircle law for the distribution of eigenvalues of specified regular subgraphs of semiregular bipartite graphs.