Unitary Cayley Graphs of Dedekind Domain Quotients
Colin Defant

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 is a commutative ring with unity, then the unitary Cayley graph of , denoted , is defined to be the graph whose vertex set is and whose edge set is . When is a Dedekind domain and is an ideal of such that is finite and nontrivial, we refer to as a \emph{generalized totient graph}. We study generalized totient graphs as generalizations of the graphs , 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 , for the number of cliques of order in a generalized totient graph. In particular, we prove that the number of cliques of…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
