Strong Resolving Graphs of Clean Graphs of Commutative Rings

TL;DR

This paper investigates the strong resolving graph and strong metric dimension of the clean graph of commutative rings, providing formulas and computations especially for Artinian rings, advancing understanding of algebraic graph structures.

Contribution

It introduces the strong resolving graph of the clean graph of a commutative ring and determines its strong metric dimension, including explicit calculations for Artinian rings.

Findings

Derived the strong resolving graph of the clean graph of a commutative ring.

Calculated the independence number of the strong resolving graph.

Determined the strong metric dimension for the clean graph of Artinian rings.

Abstract

Let $R$ be a ring with unity. The clean graph $\text{Cl}(R)$ of a ring $R$ is the simple undirected graph whose vertices are of the form $(e,u)$, where $e$ is an idempotent element and $u$ is a unit of the ring $R$ and two vertices $(e,u)$, $(f,v)$ of $\text{Cl}(R)$ are adjacent if and only if $ef = fe =0$ or $uv = vu=1$. In this manuscript, for a commutative ring $R$, first we obtain the strong resolving graph of $\text{Cl}(R)$ and its independence number. Using them, we determine the strong metric dimension of the clean graph of an arbitrary commutative ring. As an application, we compute the strong metric dimension of $\text{Cl}(R)$, where $R$ is a commutative Artinian ring.