#### Vol. 13, No. 5, 2020

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Growth series for graphs

### Walter Liu and Richard Scott

Vol. 13 (2020), No. 5, 781–790
##### Abstract

Given a graph $\Gamma$, one can associate a right-angled Coxeter group $W$ and a cube complex $\Sigma$ on which $W$ acts. By identifying $W$ with the vertex set of $\Sigma$, 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 $\Sigma$ 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.

##### Keywords
growth series, right-angled Coxeter groups, graphs
Primary: 20F55
Secondary: 51M20