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

Author

Listed:
  • Abdulaziz M. Alanazi
  • Mohd Nazim
  • Nadeem Ur Rehman
  • Li Guo

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.

Suggested Citation

  • Abdulaziz M. Alanazi & Mohd Nazim & Nadeem Ur Rehman & Li Guo, 2021. "Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings," Journal of Mathematics, Hindawi, vol. 2021, pages 1-7, August.
    • Handle: RePEc:hin:jjmath:4828579
    DOI: 10.1155/2021/4828579
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