On the domination and signed domination numbers of zero-divisor graph
<p>Let $R$ be a commutative ring (with 1) and let $Z(R)$ be its set of zero-divisors. The zero-divisor graph $\Gamma(R)$ has vertex set $Z^*(R)=Z(R) \setminus \lbrace0 \rbrace$ and for distinct $x,y \in Z^*(R)$, the vertices $x$ and $y$ are adjacent if and only if $xy=0$. In this paper, we con...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia
2016-10-01
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| Series: | Electronic Journal of Graph Theory and Applications |
| Subjects: | |
| Online Access: | https://www.ejgta.org/index.php/ejgta/article/view/83 |
| _version_ | 1818217414626639872 |
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| author | Ebrahim Vatandoost Fatemeh Ramezani |
| author_facet | Ebrahim Vatandoost Fatemeh Ramezani |
| author_sort | Ebrahim Vatandoost |
| collection | DOAJ |
| description | <p>Let $R$ be a commutative ring (with 1) and let $Z(R)$ be its set of zero-divisors. The zero-divisor graph $\Gamma(R)$ has vertex set $Z^*(R)=Z(R) \setminus \lbrace0 \rbrace$ and for distinct $x,y \in Z^*(R)$, the vertices $x$ and $y$ are adjacent if and only if $xy=0$. In this paper, we consider the domination number and signed domination number on zero-divisor graph $\Gamma(R)$ of commutative ring $R$ such that for every $0 \neq x \in Z^*(R)$, $x^2 \neq 0$. We characterize $\Gamma(R)$ whose $\gamma(\Gamma(R))+\gamma(\overline{\Gamma(R)}) \in \lbrace n+1,n,n-1 \rbrace$, where $|Z^*(R)|=n$.</p> |
| first_indexed | 2024-12-12T07:07:29Z |
| format | Article |
| id | doaj.art-1289a9a407d54a268b62115eedec6040 |
| institution | Directory Open Access Journal |
| issn | 2338-2287 |
| language | English |
| last_indexed | 2024-12-12T07:07:29Z |
| publishDate | 2016-10-01 |
| publisher | Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia |
| record_format | Article |
| series | Electronic Journal of Graph Theory and Applications |
| spelling | doaj.art-1289a9a407d54a268b62115eedec60402022-12-22T00:33:42ZengIndonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), IndonesiaElectronic Journal of Graph Theory and Applications2338-22872016-10-014214815610.5614/ejgta.2016.4.2.364On the domination and signed domination numbers of zero-divisor graphEbrahim Vatandoost0Fatemeh Ramezani1Department of Basic Science, Imam Khomeini International University, Qazvin, IranDepartment of Basic Science, Imam Khomeini International University, Qazvin, Iran<p>Let $R$ be a commutative ring (with 1) and let $Z(R)$ be its set of zero-divisors. The zero-divisor graph $\Gamma(R)$ has vertex set $Z^*(R)=Z(R) \setminus \lbrace0 \rbrace$ and for distinct $x,y \in Z^*(R)$, the vertices $x$ and $y$ are adjacent if and only if $xy=0$. In this paper, we consider the domination number and signed domination number on zero-divisor graph $\Gamma(R)$ of commutative ring $R$ such that for every $0 \neq x \in Z^*(R)$, $x^2 \neq 0$. We characterize $\Gamma(R)$ whose $\gamma(\Gamma(R))+\gamma(\overline{\Gamma(R)}) \in \lbrace n+1,n,n-1 \rbrace$, where $|Z^*(R)|=n$.</p>https://www.ejgta.org/index.php/ejgta/article/view/83domination number, signed domination number, zero-divisor graph |
| spellingShingle | Ebrahim Vatandoost Fatemeh Ramezani On the domination and signed domination numbers of zero-divisor graph Electronic Journal of Graph Theory and Applications domination number, signed domination number, zero-divisor graph |
| title | On the domination and signed domination numbers of zero-divisor graph |
| title_full | On the domination and signed domination numbers of zero-divisor graph |
| title_fullStr | On the domination and signed domination numbers of zero-divisor graph |
| title_full_unstemmed | On the domination and signed domination numbers of zero-divisor graph |
| title_short | On the domination and signed domination numbers of zero-divisor graph |
| title_sort | on the domination and signed domination numbers of zero divisor graph |
| topic | domination number, signed domination number, zero-divisor graph |
| url | https://www.ejgta.org/index.php/ejgta/article/view/83 |
| work_keys_str_mv | AT ebrahimvatandoost onthedominationandsigneddominationnumbersofzerodivisorgraph AT fatemehramezani onthedominationandsigneddominationnumbersofzerodivisorgraph |