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Geodesic growth of right-angled Coxeter groups based on trees

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.