$ A_{\alpha}  $ matrix of commuting graphs of non-abelian groups

Bilal A. Rather; Fawad Ali; Nasim Ullah; Al-Sharef Mohammad; Anwarud Din; Sehra; Bilal A. Rather; Fawad Ali; Nasim Ullah; Al-Sharef Mohammad; Anwarud Din; Sehra

doi:10.3934/math.2022845

AIMS Mathematics

2022, Volume 7, Issue 8: 15436-15452. doi: 10.3934/math.2022845

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Research article

$ A_{\alpha} $ matrix of commuting graphs of non-abelian groups

1.
Department of Mathematical Sciences, College of Science, United Arab Emirate University, Al Ain 15551, Abu Dhabi, UAE
2.
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KPK, Pakistan
3.
Department of Electrical Engineering, College of Engineering Taif University, Al-Hawiyah, Taif P.O. Box 888, Saudi Arabia
4.
Department of Mathematics, Sun Yat-Sen University, Guangzhou, China
5.
Department of Mathematics, Shaheed Benazir Bhutto Women University, Peshawar 25000, Pakistan

Received: 03 January 2022 Revised: 08 June 2022 Accepted: 16 June 2022 Published: 20 June 2022
MSC : 15A18, 05C50, 05C25

For a finite group $ \mathcal{G} $ and a subset $ X\neq \emptyset $ of $ \mathcal{G} $, the commuting graph, indicated by $ G = \mathcal{C}(\mathcal{G}, X) $, is the simple connected graph with vertex set $ X $ and two distinct vertices $ x $ and $ y $ are edge connected in $ G $ if and only if they commute in $ X $. The $ A_{\alpha} $ matrix of $ G $ is specified as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha) A(G), \; \alpha\in[0, 1] $, where $ D(G) $ is the diagonal matrix of vertex degrees while $ A(G) $ is the adjacency matrix of $ G. $ In this article, we investigate the $ A_{\alpha} $ matrix for commuting graphs of finite groups and we also find the $ A_{\alpha} $ eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest $ A_{\alpha} $ eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of $ \alpha $. Further, we show that these graphs are Laplacian integral.
- $ A_{\alpha} $ matrix,
- commuting graph,
- adjacency matrix,
- Laplacian matrix,
- signless Laplacian matrix,
- non-abelian groups
Citation: Bilal A. Rather, Fawad Ali, Nasim Ullah, Al-Sharef Mohammad, Anwarud Din, Sehra. $ A_{\alpha} $ matrix of commuting graphs of non-abelian groups[J]. AIMS Mathematics, 2022, 7(8): 15436-15452. doi: 10.3934/math.2022845

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Abstract

For a finite group $ \mathcal{G} $ and a subset $ X\neq \emptyset $ of $ \mathcal{G} $, the commuting graph, indicated by $ G = \mathcal{C}(\mathcal{G}, X) $, is the simple connected graph with vertex set $ X $ and two distinct vertices $ x $ and $ y $ are edge connected in $ G $ if and only if they commute in $ X $. The $ A_{\alpha} $ matrix of $ G $ is specified as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha) A(G), \; \alpha\in[0, 1] $, where $ D(G) $ is the diagonal matrix of vertex degrees while $ A(G) $ is the adjacency matrix of $ G. $ In this article, we investigate the $ A_{\alpha} $ matrix for commuting graphs of finite groups and we also find the $ A_{\alpha} $ eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest $ A_{\alpha} $ eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of $ \alpha $. Further, we show that these graphs are Laplacian integral.

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