Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

Abdulaziz M. Alanazi, Mohd Nazim, Nadeem Ur Rehman and Li Guo

Journal of Mathematics, 2021, vol. 2021, 1-7

Abstract: Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by ZA. An ideal ℠of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A. It is denoted by ℠≤eA. The generalized zero-divisor graph denoted by ΓgA is an undirected graph with vertex set ZA∗ (set of all nonzero zero-divisors of A) and two distinct vertices x1 and x2 are adjacent if and only if annx1+annx2≤eA. In this paper, first we characterized all the finite commutative rings A for which ΓgA is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which ΓgA is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of ΓgA.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:4828579

DOI: 10.1155/2021/4828579

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