Geodesic growth of right-angled Coxeter groups based on trees

TL;DR

This paper constructs specific families of trees to demonstrate that spectral properties of trees do not determine the geodesic growth of associated right-angled Coxeter groups, revealing limitations in spectral graph invariants.

Contribution

It provides explicit examples of non-isomorphic, co-spectral trees with identical or distinct geodesic growth in their RACGs, showing spectrum alone is insufficient to determine growth.

Findings

Constructed families of trees with same spectrum and different RACG growth

Constructed families of trees with same spectrum and same RACG growth

Proved spectrum does not determine geodesic growth asymptotically

Abstract

In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $T^1_n$ and $T^2_n$ have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree does is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $\{S^1_n\}_{n \geq 11}$ and $\{S^2_n\}_{n \geq 11}$, on $n$ vertices, such that $S^1_n$ and $S^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $S^1_n$ and $S^2_n$ have distinct geodesic growth. Asymptotically, as $n\rightarrow \infty$, each set $T^i_n$, or $S^i_n$, $i=1,2$, has the cardinality of the set of all trees on $n$ vertices. Our proofs are constructive and use two families of trees previously studied by B. McKay and C. Godsil.