Abstract
Let R be a commutative ring with non-zero identity and G be a multiplicative subgroup of U(R), where U(R) is the multiplicative group of unit elements of R. Also, suppose that S is a non-empty subset of G such that S −1={s −1∣s∈S}⫅S. Then we define Γ(R,G,S) to be the graph with vertex set R and two distinct elements x,y∈R are adjacent if and only if there exists s∈S such that x+sy∈G. This graph provides a generalization of the unit and unitary Cayley graphs. In fact, Γ(R,U(R),S) is the unit graph or the unitary Cayley graph, whenever S={1} or S={−1}, respectively. In this paper, we study the properties of the graph Γ(R,G,S) and extend some results in the unit and unitary Cayley graphs.
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Khashyarmanesh, K., Khorsandi, M.R. A generalization of the unit and unitary Cayley graphs of a commutative ring. Acta Math Hung 137, 242–253 (2012). https://doi.org/10.1007/s10474-012-0224-5
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DOI: https://doi.org/10.1007/s10474-012-0224-5