![]() ![]() 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 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. We study generalized totient graphs as generalizations of the graphs, which have appeared recently in the literature, sometimes under the name Euler totient Cayley graphs. When is a Dedekind domain and is an ideal of such that is finite and nontrivial, we refer to as a generalized totient graph. 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. ![]()
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