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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.11816 |
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| _version_ | 1866917270432776192 |
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| author | Vidya, S. Sharma, Sunny Kumar Poojary, Prasanna Alshanqiti, Omaima Bhatta, G. R. Vadiraja |
| author_facet | Vidya, S. Sharma, Sunny Kumar Poojary, Prasanna Alshanqiti, Omaima Bhatta, G. R. Vadiraja |
| contents | 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 $q>p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11816 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Metric basis and dimension of barycentric subdivision of zero divisor graphs Vidya, S. Sharma, Sunny Kumar Poojary, Prasanna Alshanqiti, Omaima Bhatta, G. R. Vadiraja Combinatorics 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 $q>p$. |
| title | Metric basis and dimension of barycentric subdivision of zero divisor graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.11816 |