Computing the strong metric dimension for co-maximal ideal graphs of commutative rings

TL;DR

This paper computes the strong metric dimension of co-maximal ideal graphs of commutative rings using Gallai's Theorem and strong resolving graphs, providing explicit formulas based on ring properties.

Contribution

It introduces a method to determine the strong metric dimension of co-maximal ideal graphs for commutative rings, with explicit formulas distinguishing reduced and non-reduced rings.

Findings

Derived explicit formulas for strong metric dimension.

Connected ring properties to graph invariants.

Applied Gallai's Theorem to algebraic graph structures.

Abstract

Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $I, J$ are adjacent if and only if $I+J=R$. In this paper, we use Gallai$^{^,}$s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.