Remarks on pseudo-vertex-transitive graphs with small diameter

Abstract

Let Γ denote a Q-polynomial distance-regular graph with vertex set X and diameter D. Let A denote the adjacency matrix of Γ. For a vertex x∈X and for 0≤i≤D, let Ei⁎(x) denote the projection matrix to the ith subconstituent space of Γ with respect to x. The Terwilliger algebra T(x) of Γ with respect to x is the semisimple subalgebra of MatX(C) generated by A,E0⁎(x),E1⁎(x),…,ED⁎(x). Let V denote a C-vector space consisting of complex column vectors with rows indexed by X. We say Γ is pseudo-vertex-transitive whenever for any vertices x,y∈X, there exists a C-vector space isomorphism ρ:V→V such that (ρA−Aρ)V=0 and (ρEi⁎(x)−Ei⁎(y)ρ)V=0 for all 0≤i≤D. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter D∈{2,3,4}. For D=2, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For D=3, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For D=4, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive.