Counting subgraphs in hyperbolic graphs with symmetry

This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generating functions, and sharp estimates on the growth rate of vertices. We then apply these results to confirm a conjecture of Saito on the "opposite series", which was originally posed for hyperbolic groups.