Growth series for graphs

Liu, Walter; Scott, Richard

doi:10.2140/involve.2020.13.781

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Growth series for graphs

Walter Liu and Richard Scott

Vol. 13 (2020), No. 5, 781–790

DOI: 10.2140/involve.2020.13.781

Abstract

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

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Keywords

growth series, right-angled Coxeter groups, graphs

Mathematical Subject Classification

Primary: 20F55

Secondary: 51M20

Milestones

Received: 17 July 2019

Revised: 12 July 2020

Accepted: 11 August 2020

Published: 5 December 2020

Communicated by Kenneth S. Berenhaut

Authors

Walter Liu
Department of Mathematics and Computer Science Santa Clara University Santa Clara, CA United States

Richard Scott
Department of Mathematics and Computer Science Santa Clara University Santa Clara, CA United States

Vol. 13, No. 5, 2020