Metric basis and dimension of barycentric subdivision of zero divisor graphs

S. Vidya; Sunny Kumar Sharma; Prasanna Poojary; Omaima Alshanqiti; G. R. Vadiraja Bhatta

paper

Metric basis and dimension of barycentric subdivision of zero divisor graphs

arXiv:2602.11816

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: $E (Γ (R)) =$ ${e = a_{1} a_{2}$ $∣$ $a_{1} \cdot a_{2} = 0$ $&$ $a_{1}, a_{2} \in V (Γ (R))}$ . 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 $q > p$ .