Remarks on pseudo-vertex-transitive graphs with small diameter

Jack H. Koolen, Jae-Ho Lee, Ying-Ying Tan

Published 2021-01-29Version 1

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. Fix a base vertex $x\in X$ and for $0 \leq i \leq D$ let $E^*_i=E^*_i(x)$ denote the projection matrix to the $i$th subconstituent space of $\Gamma$ with respect to $x$. The Terwilliger algebra $T(x)$ of $\Gamma$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0, E^*_1, \ldots, E^*_D$. Remark that the isomorphism class of $T(x)$ depends on the choice of the base vertex $x$. We say $\Gamma$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, the Terwilliger algebras $T(x)$ and $T(y)$ are isomorphic. In this paper we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. In the case of diameter $2$, a strongly regular graph $\Gamma$ is thin, and $\Gamma$ is pseudo-vertex-transitive if and only if every local graph of $\Gamma$ has the same spectrum. In the case of diameter $3$, Taylor graphs are thin and pseudo-vertex-transitive. In the case of diameter $4$, antipodal tight graphs are thin and pseudo-vertex-transitive.