Walks on Unitary Cayley Graphs and Applications

In this paper, we determine an explicit formula for the number of walks in $X_n=\mathsf{\text{Cay}}({\mathbb{Z}}_n,{\mathbb{U}}_n)$, 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 $n$ as the sum of $k$ units of ${\mathbb{Z}}_n$.