# Nil clean graph of rings

## TL;DR

This paper introduces the nil clean graph of a ring, exploring its properties such as girth, dominating set, and diameter specifically for finite commutative rings, linking algebraic and graph-theoretic concepts.

## Contribution

It defines the nil clean graph of a ring and investigates its fundamental graph properties within the context of finite commutative rings.

## Key findings

- Analyzed girth, diameter, and dominating sets of nil clean graphs.
- Established relationships between ring properties and graph structure.
- Provided new insights into the interplay between ring theory and graph theory.

## Abstract

In this article, we have defined nil clean graph of a ring $R$. The vertex set is the ring $R$, two ring elements $a$ and $b$ are adjacent if and only if $a + b$ is nil clean in $R$. Graph theoretic properties like girth, dominating set, diameter etc. of nil clean graph have been studied for finite commutative rings.