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

Abstract

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