Unitary Cayley Graphs of Dedekind Domain Quotients

TL;DR

This paper introduces generalized totient graphs derived from Dedekind domain quotients, providing formulas for clique counts, and analyzing various graph properties including chromatic and domination numbers, correcting previous misconceptions.

Contribution

It extends the concept of Euler totient Cayley graphs to Dedekind domains, introduces a Schemmel totient function generalization, and analyzes key graph invariants.

Findings

Derived a formula for the number of cliques in generalized totient graphs.

Determined properties like clique number, chromatic number, and girth for these graphs.

Corrected previous errors regarding domination numbers in Euler totient Cayley graphs.

Abstract

If $X$ is a commutative ring with unity, then the unitary Cayley graph of $X$, denoted $G_X$, is defined to be the graph whose vertex set is $X$ and whose edge set is $\{\{a,b\}\colon a-b\in X^\times\}$. When $R$ is a Dedekind domain and $I$ is an ideal of $R$ such that $R/I$ is finite and nontrivial, we refer to $G_{R/I}$ as a \emph{generalized totient graph}. We study generalized totient graphs as generalizations of the graphs $G_{\mathbb{Z}/(n)}$, which have appeared recently in the literature, sometimes under the name \emph{Euler totient Cayley graphs}. We begin by generalizing to Dedekind domains the arithmetic functions known as Schemmel totient functions, and we use one of these generalizations to provide a simple formula, for any positive integer $m$, for the number of cliques of order $m$ in a generalized totient graph. In particular, we prove that the number of cliques of order $m$ in $G_{\mathbb Z/(n)}$ is \[\prod_{k=1}^m\frac{S_{k-1}(n)}{k},\] where $S_r$ is the $r^{\text{th}}$ Schemmel totient function. We then proceed to determine many properties of generalized totient graphs such as their clique numbers, chromatic numbers, chromatic indices, clique domination numbers, and (in many, but not all cases) girths. We also determine the diameter of each component of a generalized totient graph. We correct one erroneous claim about the clique domination numbers of Euler totient Cayley graphs that has appeared in the literature and provide a counterexample to a second claim about the strong domination numbers of these graphs.