The adjacency matrices and the transition matrices related to random walks on graphs

Abstract

A pointed graph (Γ, v 0 ) induces a family of transition matrices in Wildberger's construction of a hermitian hypergroup via a random walk on Γ starting from v 0 . We will give a necessary condition for producing a hermitian hypergroup as we assume a weaker condition than the distance-regularity for (Γ, v 0 ). The condition obtained in this paper connects the transition matrices and the adjacency matrices associated with Γ.