Metric basis and dimension of barycentric subdivision of zero divisor graphs
S. Vidya, Sunny Kumar Sharma, Prasanna Poojary, Omaima Alshanqiti, G. R. Vadiraja Bhatta
TL;DR
This work analyzes the metric dimension of the barycentric subdivision of the zero-divisor graph $\Gamma(\mathbb{Z}_{pq})$ for distinct primes $p<q$. It develops resolving-set constructions and upper/lower bound arguments to establish $dim(BS(\Gamma(\mathbb{Z}_{pq})))=q-2$ when $q \ge 2p-1$, with a noted boundary case $q=2p-3$ yielding $dim(BS(\Gamma(\mathbb{Z}_{pq})))=q-1$. The approach extends to the small-prime scenarios $n=2q$ and $n=3q$, where the same bound holds, and frames the results within the broader study of metric dimensions under barycentric subdivision. The findings contribute to the theory of graph transformations linked to zero-divisor graphs and have potential implications for network design and combinatorial topology where subdivision-based metrics are relevant.
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 $q>p$.