Metric basis and dimension of barycentric subdivision of zero divisor graphs

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 $\Gamma(R) = (V(\Gamma(R)), E(\Gamma(R)))$, where the vertex set $V(\Gamma(R))$ consists of all the non-zero zero-divisors of $R$, and the edge set $E(\Gamma(R))$ is defined as follows: $E(\Gamma(R)) =$ $\{e = a_1a_2$ $|$ $a_1 \cdot a_2 = 0$ $\&$ $a_1, a_2 \in V(\Gamma(R))\}$. In this article, we consider the zero divisor graph of a group of integers modulo $n$, denoted as $\Gamma(\mathbb{Z}_n)$, 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 $\Gamma(\mathbb{Z}_n)$, denoted by $dim(BS(\Gamma(\mathbb{Z}_n)))$, and we also prove that $dim(BS(\Gamma(\mathbb{Z}_n)))\geq q-2$ for every $n=pq$, where $p$ and $q$ are distinct primes and $q>p$.