Walks on Unitary Cayley Graphs and Applications
  In this paper, we determine an explicit formula for the number of walks in $X_n = \textsf{Cay}(\mathbb{Z}_n,\mathbb{U}_n)$, 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 $\bmod{}n$ as the sum of $k$ units of $\mathbb{Z}_n$. 
Elias Cancela, Daniel A. Jaume, Adri\'an Pastine, Denis Videla
math.CO math.NT