Enumerating Cliques in Direct Product Graphs

TL;DR

This paper explores the enumeration of cliques in unitary Cayley graphs and their generalizations, providing formulas involving Schemmel totient functions for counting cliques of any size.

Contribution

It introduces a simple formula for counting cliques in unitary Cayley graphs and extends this to direct products of balanced complete multipartite graphs.

Findings

Derived a formula for the number of cliques in unitary Cayley graphs.

Connected clique enumeration to Schemmel totient functions.

Extended clique counting formulas to broader graph classes.

Abstract

The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erd\H{o}s and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order $m$ in a direct product of balanced complete multipartite graphs.