Remarks on pseudo-vertex-transitive graphs with small diameter

Let Gamma denote a Q-polynomial distance-regular graph with vertex set X and diameter D. Let A denote the adjacency matrix of Gamma. For a vertex x is an element of X and for 0 <= i <= D, let E-i*(x) denote the projection matrix to the ith subconstituent space of Gamma with respect to x. The Terwilliger algebra T (x) of Gamma with respect to x is the semisimple subalgebra of Mat(X)(C) generated by A, E-0*(x), E-1*(x), ... ,E-D*(x). Let V denote a C-vector space consisting of complex column vectors with rows indexed by X. We say Gamma is pseudo-vertex-transitive whenever for any vertices x, y e X, there exists a C-vector space isomorphism rho : V -> V such that (rho A -A rho)V = 0 and (rho E-i*(x) - E-i* (y)rho)V = 0 for all 0 <= i <= D. In this paper, we discuss pseudo vertex transitivity for distance-regular graphs with diameter D is an element of{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. (c) 2022 Elsevier B.V. All rights reserved.

Automated summary

This paper discusses the pseudo-vertex transitivity of distance-regular graphs with diameters 2, 3, and 4. The results show that a strongly regular graph is pseudo-vertex transitive if and only if its local graphs have the same spectrum. Additionally, it is shown that Taylor graphs and antipodal tight graphs are pseudo-vertex transitive.