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.