Fault-tolerant metric basis and dimension of barycentric subdivision of zero divisor graphs

TL;DR

This paper investigates the fault-tolerant metric dimension of the barycentric subdivision of zero divisor graphs for integers modulo n, providing bounds and properties for graphs derived from products of two distinct odd primes.

Contribution

It introduces the concept of fault-tolerant metric dimension for barycentric subdivisions of zero divisor graphs and establishes a lower bound for this dimension when n is a product of two distinct odd primes.

Findings

The fault-tolerant metric dimension is at least q - 1 for n = pq.

The study focuses on zero divisor graphs of the group of integers modulo n.

Provides bounds for the fault-tolerant metric dimension of these graphs.

Abstract

The undirected zero divisor graph of a commutative ring with unity \( R \), denoted by \( \Gamma(R) = (V(\Gamma(R)), E(\Gamma(R))) \). The vertex set \( V(\Gamma(R)) \) consists of all the non-zero zero-divisors of \( R \). The edge set \( E(\Gamma(R)) \) is defined by the set \( \{ e = a_1 a_2 \mid a_1 \cdot a_2 = 0 \text{ and } a_1, a_2 \in V(\Gamma(R)) \} \). The barycentric subdivision of $\Gamma$ is the process of subdividing each edge by inserting new vertex in the graph $\Gamma$. In this article, we have focused on the fault-tolerant metric dimension of the barycentric subdivision of zero divisor graph of the group of integers modulo \( n \), represented by \( fdim(BS(\Gamma(\mathbb{Z}_n )\), where \( n = pq \); \( p \) and \( q \) are distinct odd primes with \( q > p \). We also demonstrate that \( fdim(BS(\Gamma(\mathbb{Z}_n) \geq q - 1 \) for every \( n = pq \), where \( p \) and \( q \) are any distinct odd primes with \( q > p \).