On the offensive alliance number for the zero divisor graph of $ \mathbb{Z}_n $

José Ángel Juárez Morales; Jesús Romero Valencia; Raúl Juárez Morales; Gerardo Reyna Hernández; José Ángel Juárez Morales; Jesús Romero Valencia; Raúl Juárez Morales; Gerardo Reyna Hernández

doi:10.3934/mbe.2023539

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 12118-12129. doi: 10.3934/mbe.2023539

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On the offensive alliance number for the zero divisor graph of $ \mathbb{Z}_n $

1.
Faculty of Mathematics, Autonomous University of Guerrero, Acapulco, Guerrero, Mexico
2.
Faculty of Mathematics, Autonomous University of Guerrero, Chilpancingo, Mexico
3.
Faculty of Science and information Technology, Autonomous University of Guerrero, Acapulco, Guerrero, México

Received: 21 March 2023 Revised: 20 April 2023 Accepted: 26 April 2023 Published: 15 May 2023

A nonempty subset $ D $ of vertices in a graph $ \Gamma = (V, E) $ is said is an offensive alliance, if every vertex $ v \in \partial(D) $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum offensive alliance of $ \Gamma $ is called the offensive alliance number $ \alpha ^o(\Gamma) $ of $ \Gamma $. An offensive alliance $ D $ is called global, if every $ v \in V - D $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum global offensive alliance of $ \Gamma $ is called the global offensive alliance number $ \gamma^o(\Gamma) $ of $ \Gamma $. For a finite commutative ring with identity $ R $, $ \Gamma(R) $ denotes the zero divisor graph of $ R $. In this paper, we compute the offensive alliance (global, independent, and independent global) numbers of $ \Gamma(\mathbb{Z}_n) $, for some cases of $ n $.
- Offensive alliance,
- global offensive alliance,
- alliance number,
- Zero divisor graph,
- independent offensive alliance
Citation: José Ángel Juárez Morales, Jesús Romero Valencia, Raúl Juárez Morales, Gerardo Reyna Hernández. On the offensive alliance number for the zero divisor graph of $ \mathbb{Z}_n $[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 12118-12129. doi: 10.3934/mbe.2023539

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Abstract

A nonempty subset $ D $ of vertices in a graph $ \Gamma = (V, E) $ is said is an offensive alliance, if every vertex $ v \in \partial(D) $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum offensive alliance of $ \Gamma $ is called the offensive alliance number $ \alpha ^o(\Gamma) $ of $ \Gamma $. An offensive alliance $ D $ is called global, if every $ v \in V - D $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum global offensive alliance of $ \Gamma $ is called the global offensive alliance number $ \gamma^o(\Gamma) $ of $ \Gamma $. For a finite commutative ring with identity $ R $, $ \Gamma(R) $ denotes the zero divisor graph of $ R $. In this paper, we compute the offensive alliance (global, independent, and independent global) numbers of $ \Gamma(\mathbb{Z}_n) $, for some cases of $ n $.

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