Walks on Unitary Cayley Graphs and Applications

TL;DR

This paper derives an explicit formula for counting walks in unitary Cayley graphs of integers modulo n, enabling analysis of sum representations of residue classes as sums of units.

Contribution

It provides a new explicit formula for walks in unitary Cayley graphs and applies it to count representations of residue classes as sums of units.

Findings

Explicit formula for walks in $X_n$

Counts of residue class representations as sums of units

Applications to number theory problems

Abstract

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 $\mathbb{Z}_n$.