Nil Clean Divisor Graph

TL;DR

This paper introduces the nil clean divisor graph, a new graph structure based on finite commutative rings, exploring its properties and relationships with nil clean elements.

Contribution

It defines the nil clean divisor graph for finite commutative rings and investigates its properties, providing new insights into the interplay between ring elements and graph theory.

Findings

Characterization of vertices in the nil clean divisor graph

Conditions for adjacency based on nil clean products

Structural properties of the nil clean divisor graph

Abstract

In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\, x\neq 0, \,\exists\, y(\neq 0, \neq x)\in R$ such that $xy$ is nil clean$\}$, two vertices $x$ and $y$ are adjacent if $xy$ is a nil clean element. We prove some interesting results of nil clean divisor graph of a ring.