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 $\Gamma(R) = (V(\Gamma(R)), E(\Gamma(R)))$, 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 $q>p$.