A novel edge-weighted matrix of a graph and its spectral properties with potential applications

Sakander Hayat; Sunilkumar M. Hosamani; Asad Khan; Ravishankar L. Hutagi; Umesh S. Mujumdar; Mohammed J. F. Alenazi; Sakander Hayat; Sunilkumar M. Hosamani; Asad Khan; Ravishankar L. Hutagi; Umesh S. Mujumdar; Mohammed J. F. Alenazi

doi:10.3934/math.20241216

AIMS Mathematics

2024, Volume 9, Issue 9: 24955-24976. doi: 10.3934/math.20241216

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A novel edge-weighted matrix of a graph and its spectral properties with potential applications

1.
Mathematical Sciences, Faculty of Science, Universiti Brunei Darussalam, Jln Tungku Link, Gadong BE1410, Brunei Darussalam
2.
Department of Mathematics, Rani Channamma University's Sangolli Rayanna First Grade Constituent College, Belagavi, 590001, India
3.
Metaverse Research Institute, School of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou, Guangdong 510006, China
4.
Department of Computer Engineering, College of Computer and Information Sciences (CCIS), King Saud University, Riyadh 11451, Saudi Arabia

Received: 10 June 2024 Revised: 16 July 2024 Accepted: 24 July 2024 Published: 27 August 2024
MSC : 05C12, 05C50, 05C92

Regarding a simple graph $ \Gamma $ possessing $ \nu $ vertices ($ \nu $-vertex graph) and $ m $ edges, the vertex-weight and weight of an edge $ e = uv $ are defined as $ w(v_{i}) = d_{ \Gamma}(v_{i}) $ and $ w(e) = d_{ \Gamma}(u)+d_{ \Gamma}(v)-2 $, where $ d_{ \Gamma}(v) $ is the degree of $ v $. This paper puts forward a novel graphical matrix named the edge-weighted adjacency matrix (adjacency of the vertices) $ A_{w}(\Gamma) $ of a graph $ \Gamma $ and is defined in such a way that, for any $ v_{i} $ that is adjacent to $ v_{j} $, its $ (i, j) $-entry equals $ w(e) = d_{ \Gamma}(v_{i})+d_{ \Gamma}(v_{j})-2 $; otherwise, it equals 0. The eigenvalues $ \lambda_{1}^{w}\ge \lambda_{2}^{w}\ge\ldots\ge \lambda_{\nu}^{w} $ of $ A_w $ are called the edge-weighted eigenvalues of $ \Gamma $. We investigate the mathematical properties of $ A_{w}(\Gamma) $'s spectral radius $ \lambda_{1}^{w} $ and energy $ E_{w}(\Gamma) = \sum_{i = 1}^{\nu}|\lambda_{i}^{w}| $. Sharp lower and upper bounds are obtained for $ \lambda_{1}^{w} $ and $ E_{w}(\Gamma) $, and the respective extremal graphs are characterized. Further, we employ these spectral descriptors in structure-property modeling of the physicochemical properties of polycyclic aromatic hydrocarbons for a set of benzenoid hydrocarbons (BHs). Detailed regression analysis showcases that edge-weighted energy outperforms classical adjacency energy in structure-property modeling of the physicochemical properties of BHs.
- edge weight energy,
- spectral radius,
- benzenoid hydrocarbons,
- structure-property modeling,
- graphical descriptor
Citation: Sakander Hayat, Sunilkumar M. Hosamani, Asad Khan, Ravishankar L. Hutagi, Umesh S. Mujumdar, Mohammed J. F. Alenazi. A novel edge-weighted matrix of a graph and its spectral properties with potential applications[J]. AIMS Mathematics, 2024, 9(9): 24955-24976. doi: 10.3934/math.20241216

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Abstract

Regarding a simple graph $ \Gamma $ possessing $ \nu $ vertices ($ \nu $-vertex graph) and $ m $ edges, the vertex-weight and weight of an edge $ e = uv $ are defined as $ w(v_{i}) = d_{ \Gamma}(v_{i}) $ and $ w(e) = d_{ \Gamma}(u)+d_{ \Gamma}(v)-2 $, where $ d_{ \Gamma}(v) $ is the degree of $ v $. This paper puts forward a novel graphical matrix named the edge-weighted adjacency matrix (adjacency of the vertices) $ A_{w}(\Gamma) $ of a graph $ \Gamma $ and is defined in such a way that, for any $ v_{i} $ that is adjacent to $ v_{j} $, its $ (i, j) $-entry equals $ w(e) = d_{ \Gamma}(v_{i})+d_{ \Gamma}(v_{j})-2 $; otherwise, it equals 0. The eigenvalues $ \lambda_{1}^{w}\ge \lambda_{2}^{w}\ge\ldots\ge \lambda_{\nu}^{w} $ of $ A_w $ are called the edge-weighted eigenvalues of $ \Gamma $. We investigate the mathematical properties of $ A_{w}(\Gamma) $'s spectral radius $ \lambda_{1}^{w} $ and energy $ E_{w}(\Gamma) = \sum_{i = 1}^{\nu}|\lambda_{i}^{w}| $. Sharp lower and upper bounds are obtained for $ \lambda_{1}^{w} $ and $ E_{w}(\Gamma) $, and the respective extremal graphs are characterized. Further, we employ these spectral descriptors in structure-property modeling of the physicochemical properties of polycyclic aromatic hydrocarbons for a set of benzenoid hydrocarbons (BHs). Detailed regression analysis showcases that edge-weighted energy outperforms classical adjacency energy in structure-property modeling of the physicochemical properties of BHs.

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