Metric basis and dimension of barycentric subdivision of zero divisor graphs

Abstract: Let $R$ be a commutative ring with unity 1, and $ G(V,E)$ be a simple, connected, nontrivial graph. Let $d(a,c)$ be the distance between the vertices $a$ and $c $ in $G$. An undirected zero divisor graph of a ring $R$ is denoted by $Γ(R) = (V(Γ(R)), E(Γ(R)))$, where the vertex set $V(Γ(R))$ consists of all the non-zero zero-divisors of $R$, and the edge set $E(Γ(R))$ is defined as follows: $E(Γ(R)) = $ ${e = a_1a_2$ $ |$ $ a_1 \cdot a_2 = 0$ $&$ $ a_1, a_2 \in V(Γ(R))}$. In this article, we consider the zero divisor graph of a group of integers modulo (n), denoted as (Γ(\mathbb{Z}_n)), where (n=pq). Here, (p) and (q) are distinct primes, with (q > p). We aim to determine the metric dimension of the barycentric subdivision of the zero divisor graph (Γ(\mathbb{Z}_n)), denoted by (dim(BS(Γ(\mathbb{Z}_n)))), and we also prove that (dim(BS(Γ(\mathbb{Z}_n)))\geq q-2) for every (n=pq), where (p) and (q) are distinct primes and $q>p$.