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

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