On the planarity, genus, and crosscap of the weakly zero-divisor graph of commutative rings

• Nadeem ur Rehman ^[1] ; Mohd Nazim ^[1] ; Shabir Ahmad Mir ^[1]
  1. [1] Aligarh Muslim University

     Aligarh Muslim University

     India

     
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 67, Nº. 1, 2024, págs. 213-227
• Idioma: inglés
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• Resumen

  • Let R be a commutative ring and Z(R) its zero-divisors set. The weakly zero-divisor graph of R, denoted by WΓ(R) is an undirected graph with the nonzero zero-divisors Z(R)∗as vertex set and two distinct vertices x and y are adjacent if and only if there exist a∈Ann(x) and b∈Ann(y) such that ab=0. In this paper, we characterize finite rings R for which the weakly zero-divisor graph WΓ(R) belongs to some well-known families of graphs. Further, we classify the finite rings R for which WΓ(R) is planar, toroidal or double toroidal. Finally, we classify the finite rings R for which the graph WΓ(R) has crosscap at most two.