On the strong metric dimension of the zero-divisor graph of a lattice

TL;DR

This paper introduces a generalized blow-up of Boolean lattices and computes the strong metric dimension of their zero-divisor graphs, with applications to various algebraic and graph-theoretic structures.

Contribution

It presents a novel method to determine the strong metric dimension of zero-divisor graphs of lattice blow-ups and applies it to multiple algebraic graphs.

Findings

Calculated the strong metric dimension of zero-divisor graphs of Boolean lattice blow-ups.

Extended results to comaximal and ideal graphs, and zero-divisor graphs of reduced rings.

Provided formulas and methods for these computations.

Abstract

In this paper, the generalized blow-up of a Boolean lattice $L\cong \textbf{2}^n$ using finite chains is introduced. Also, we compute the strong metric dimension of the zero-divisor graph of the blow-up of a Boolean lattice. These results are applied to calculate the strong metric dimension of the comaximal graph, the comaximal ideal graph, the zero-divisor graph of a reduced ring, and the component graph of a vector space.