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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Kroschel, Brenda K.
Department of Mathematics, University of St. Thomas, St. Paul, MN, USA
Electrical & Electronics Engineering

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On domination numbers of zero-divisor graphs of commutative rings Anderson, Sarah E.; Axtell, Mike; Kroschel, Brenda K.; Stickles, Jr., Joe A.
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 12, No 2 (2024): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2024.12.2.2

Abstract

Zero-divisor graphs of a commutative ring R, denoted Γ(R), are well-represented in the literature. In this paper, we consider domination numbers of zero-divisor graphs. For reduced rings, Vatandoost and Ramezani characterized the possible graphs for Γ(R) when the sum of the domination numbers of Γ(R) and the complement of Γ(R) is n - 1, n, and n + 1, where n is the number of nonzero zero-divisors of R. We extend their results to nonreduced rings, determine which graphs are realizable as zero-divisor graphs, and provide the rings that yield these graphs.
Co-Authors Anderson, Sarah E. Axtell, Mike Stickles, Jr., Joe A.