On the Walks on Cayley Graphs‎

Abstract

‎Let $G$ be a group and $S$ be an inverse-closed subset of $G$ not contining of the identity element of $G$‎. ‎The Cayley graph‎‎of $G$ with respect to $S$‎, ‎$\Cay(G,S)$‎, ‎is a graph with vertex set $G$ and edge set $\{\{g,sg\}\mid g\in G,s\in S\}$‎.‎In this paper‎, ‎we compute the number of walks of any length between two arbitrary vertices of $\Cay(G,S)$ in terms‎‎of complex irreducible representations of $G$‎. ‎Using our main result‎, ‎we give exact formulas for the number of walks‎‎of any length between two vertices in complete graphs‎, ‎cycles‎, ‎complete bipartite graphs‎, ‎Hamming graphs and complete transposition‎‎graphs‎.