Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring

TL;DR

This paper studies the essential ideal graph of a finite commutative ring, characterizing its metric dimension, isomorphisms with other graphs, and calculating Zagreb indices, thereby advancing algebraic graph theory.

Contribution

It introduces the concept of metric dimension for the essential ideal graph and establishes isomorphisms with annihilating ideal graphs for certain rings, along with calculating topological indices.

Findings

The essential ideal graph has finite metric dimension.

The essential ideal graph of Z_n is isomorphic to its annihilating ideal graph when n is a product of distinct primes.

The Zagreb indices of the essential ideal graph are explicitly determined.

Abstract

Let $R$ be a commutative ring with unity. The essential ideal graph $\mathcal{E}_{R}$ of $R$ is a graph whose vertex set consists of all nonzero proper ideals of \textit{R}. Two vertices $\hat{I}$ and $\hat{J}$ are adjacent if and only if $\hat{I}+ \hat{J}$ is an essential ideal. In this paper, we characterize the graph $\mathcal{E}_{R}$ as having a finite metric dimension. Additionally, we identify that the essential ideal graph and annihilating ideal graph of the ring $\mathbb{Z}_{n}$ are isomorphic whenever $n$ is a product of distinct primes. Also, we estimate the metric dimension of the essential ideal graph of the ring $\mathbb{Z}_{n}$. Furthermore, we determine the topological indices, namely the first and the second Zagreb indices, of $\mathcal{E}_{\mathbb Z_n}$.