Geodesic Growth of Numbered Graph Products

TL;DR

This paper investigates the geodesic growth of numbered graph products, a generalization of right-angled Coxeter groups, introducing link-regularity and deriving formulas for their growth series.

Contribution

It introduces link-regularity as a key condition and provides formulas and systems of equations to compute geodesic growth for these groups.

Findings

Link-regular numbered graphs determine identical geodesic growth series.

Derived explicit formulas for geodesic growth of right-angled Coxeter groups.

Established a solvable system of equations for specific graph classes.

Abstract

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.