Walks on Unitary Cayley Graphs and Applications

Abstract

In this paper, we determine an explicit formula for the number of walks in $X_n=\text{Cay}({\mathbb{Z}}_n,{\mathbb{U}}_n)$, the unitary Cayley Graphs of order $n$, between any pair of its vertices. With this result, we give the number of representations of a fixed residue class $\mathrm{mod}n$ as the sum of $k$ units of ${\mathbb{Z}}_n$.