  In this paper, we study geodesic growth of numbered graph products; these are
a generalization of right-angled Coxeter groups, defined as graph products of
finite cyclic groups. We first define a graph-theoretic condition called
link-regularity, as well as a natural equivalence amongst link-regular numbered
graphs, and show that numbered graph products associated to link-regular
numbered graphs must have the same geodesic growth series. Next, we derive a
formula for the geodesic growth of right-angled Coxeter groups associated to
link-regular graphs. Finally, we find a system of equations that can be used to
solve for the geodesic growth of numbered graph products corresponding to
link-regular numbered graphs that contain no triangles and have constant vertex
numbering.
