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 $\mathrm{\Gamma }(R)=(V(\mathrm{\Gamma }(R)),E(\mathrm{\Gamma }(R)))$, where the vertex set $V(\mathrm{\Gamma }(R))$ consists of all the non-zero zero-divisors of $R$, and the edge set $E(\mathrm{\Gamma }(R))$ is defined as follows: $E(\mathrm{\Gamma }(R))=$ $\{e=a_1{a}_2$ $\mathrm{\mid }$ $a_1\cdot a_2=0$ $\mathrm{\&}$ $a_1,a_2\in V(\mathrm{\Gamma }(R))\}$. In this article, we consider the zero divisor graph of a group of integers modulo $n$, denoted as $\mathrm{\Gamma }({\mathbb{Z}}_n)$, 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 $\mathrm{\Gamma }({\mathbb{Z}}_n)$, denoted by $dim(BS(\mathrm{\Gamma }({\mathbb{Z}}_n)))$, and we also prove that $dim(BS(\mathrm{\Gamma }({\mathbb{Z}}_n)))\ge q-2$ for every $n=pq$, where $p$ and $q$ are distinct primes and $q>p$.