On a new extension of the zero-divisor graph (II)

TL;DR

This paper extends the study of a new zero-divisor graph, characterizing its diameters for rings and polynomial rings, and exploring relationships with other graph structures related to zero-divisors.

Contribution

It provides a complete characterization of diameters of the extended zero-divisor graph for rings and polynomial rings, and examines its relations with other zero-divisor related graphs.

Findings

Characterization of possible diameters of the extended zero-divisor graph.

Analysis of the relationship between zero-divisor graph, total graph subgraph, and extended graph.

Identification of properties of the extended zero-divisor graph.

Abstract

We continue our study of the new extension of zero-divisor graph. We give a complete characterization for the possible diameters of $\widetilde{\Gamma}(R)$ and $\widetilde{\Gamma}(R[x_1,\dots,x_n])$, we investigate the relation between the zero-divisor graph, the subgraph of total graph on $Z(R)^{\star}$ and $\widetilde{\Gamma}(R)$ and we present some other properties of $\widetilde{\Gamma}(R)$.