  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.

Group Theory;Geometric Topology