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

TL;DR

This paper explores the relationship between adjacency and transition matrices in graphs, providing a necessary condition for constructing hermitian hypergroups from random walks on graphs with weaker assumptions than distance-regularity.

Contribution

It introduces a new necessary condition linking adjacency and transition matrices for hermitian hypergroup construction, relaxing previous distance-regularity assumptions.

Findings

Established a necessary condition for hermitian hypergroup formation

Connected transition matrices with adjacency matrices under weaker conditions

Extended understanding of random walks on graphs in hypergroup context

Abstract

A pointed graph $(\Gamma,v_0)$ induces a family of transition matrices in Wildberger's construction of a hermitian hypergroup via a random walk on $\Gamma$ 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 $(\Gamma,v_0)$. The condition obtained in this paper connects the transition matrices and the adjacency matrices associated with $\Gamma$.