Domination in generalized unit and unitary Cayley graphs of finite rings

T. Tamizh Chelvam (), S. Anukumar Kathirvel () and M. Balamurugan ()
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T. Tamizh Chelvam: Manonmaniam Sundaranar University
S. Anukumar Kathirvel: Manonmaniam Sundaranar University
M. Balamurugan: Manonmaniam Sundaranar University

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 2, 533-556

Abstract: Abstract Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph Γ is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that x + uy is a unit in R. Also, $$\overline{\Gamma}$$Γ¯ denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as $$\overline{\Gamma}$$Γ¯ and characterize all γ-sets in Γ and $$\overline{\Gamma}$$Γ¯. Also, we obtain the bondage number of Γ. Further, we obtain the values of some domination parameters like independent, strong and weak domination numbers of $$\overline{\Gamma}$$Γ¯.

Keywords: Commutative rings; generalized unit and unitary graph; Cayley graphs; complement graph; domination number; independent number; 05C25; 05C69; 16P10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13226-020-0415-7

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