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

Abstract

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z A . An ideal ℐ 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 ℐ ≤ e A . The generalized zero-divisor graph denoted by Γ g A is an undirected graph with vertex set Z A ∗ (set of all nonzero zero-divisors of A ) and two distinct vertices x 1 and x 2 are adjacent if and only if ann x 1 + ann x 2 ≤ e A . In this paper, first we characterized all the finite commutative rings A for which Γ g A is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Γ g A is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Γ g A .

CONSTRUCT THE TRIPLE ZERO GRAPH OF RING Z_n USING PYTHON

“…Such a graph is denoted by Γ(𝑅). The concept of a zero divisor graph of a commutative ring according to has prompted much research afterward, as in [5], [6], [7], [8], and [9].…”

Connectedness in Projection Graphs of Commutative Rings

“…During last thirty years, many authors published research articles on zero-divisor graphs in different forms. To mention some of them, Abdulaziz et al characterized rings using generalized zero-divisor graph (1) . In 2021 Vijay Kumar Bhat considered matrix ring of order n over Z p and associated zero-divisor graph (2) ; Pradeep Singh et al investigated connectivity (3) ; Sriparna Chattopadhyay et al studied Laplacian eigenvalues (4) .…”