Growth series for graphs

Abstract

Given a graph $\mathrm{\Gamma }$, one can associate a right-angled Coxeter group $W$ and a cube complex $\mathrm{\Sigma }$ on which $W$ acts. By identifying $W$ with the vertex set of $\mathrm{\Sigma }$, one obtains a growth series for $W$ defined as $W\left(t\right)={\sum }_{w\in W}{t}^{\ell \left(w\right)}$, where $\ell \left(w\right)$ denotes the minimum length of an edge path in $\mathrm{\Sigma }$ from the vertex $1$ to the vertex $w$. The series $W\left(t\right)$ is known to be a rational function. We compute some examples and investigate the poles and zeros of this function.