Complemented zero-divisor graph of posets
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908973268992000 |
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| author | Khiste, Anagha Tarte, Ganesh Joshi, Vinayak |
| author_facet | Khiste, Anagha Tarte, Ganesh Joshi, Vinayak |
| contents | In this paper, we derive a set of equivalent conditions for the zero-divisor graph $Γ(Q)$ of a poset $Q$ with $0$ to be complemented, characterizing it in terms of quasi-complemented posets. Furthermore, we prove that the notions of a complemented zero-divisor graph and a uniquely complemented zero-divisor graph coincide for any poset $Q$ with $0$. In addition, we provide both algebraic and topological characterizations for $Γ(Q)$ to be a complemented graph. In the final section, we apply these characterizations to the zero-divisor graphs of a reduced (multiplicative) semigroup $S$ with $0$ and the comaximal (ideal) graph of an Artinian ring $R$, and the nonzero component union graph $\mathbb{UG}(\mathbb{V})$ of a finite-dimensional vector space $\mathbb{V}$ over a field $\mathbb{F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15498 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complemented zero-divisor graph of posets Khiste, Anagha Tarte, Ganesh Joshi, Vinayak Combinatorics Primary 05C15, Secondary 06A12, 13A70 In this paper, we derive a set of equivalent conditions for the zero-divisor graph $Γ(Q)$ of a poset $Q$ with $0$ to be complemented, characterizing it in terms of quasi-complemented posets. Furthermore, we prove that the notions of a complemented zero-divisor graph and a uniquely complemented zero-divisor graph coincide for any poset $Q$ with $0$. In addition, we provide both algebraic and topological characterizations for $Γ(Q)$ to be a complemented graph. In the final section, we apply these characterizations to the zero-divisor graphs of a reduced (multiplicative) semigroup $S$ with $0$ and the comaximal (ideal) graph of an Artinian ring $R$, and the nonzero component union graph $\mathbb{UG}(\mathbb{V})$ of a finite-dimensional vector space $\mathbb{V}$ over a field $\mathbb{F}$. |
| title | Complemented zero-divisor graph of posets |
| topic | Combinatorics Primary 05C15, Secondary 06A12, 13A70 |
| url | https://arxiv.org/abs/2604.15498 |