On the general atom-bond sum-connectivity index

Abeer M. Albalahi; Zhibin Du; Akbar Ali; Abeer M. Albalahi; Zhibin Du; Akbar Ali

doi:10.3934/math.20231210

AIMS Mathematics

2023, Volume 8, Issue 10: 23771-23785. doi: 10.3934/math.20231210

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On the general atom-bond sum-connectivity index

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il, Saudi Arabia
2.
School of Software, South China Normal University, Foshan, China

Received: 20 May 2023 Revised: 13 July 2023 Accepted: 25 July 2023 Published: 03 August 2023
MSC : 05C07, 05C90

This paper is concerned with a generalization of the atom-bond sum-connectivity (ABS) index, devised recently in [A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081-2093]. For a connected graph $ G $ with an order greater than $ 2 $, the general atom-bond sum-connectivity index is represented as $ ABS_\gamma(G) $ and is defined as the sum of the quantities $ (1-2(d_x+d_y)^{-1})^{\gamma} $ over all edges $ xy $ of the graph $ G $, where $ d_x $ and $ d_y $ represent the degrees of the vertices $ x $ and $ y $ of $ G $, respectively, and $ \gamma $ is any real number. For $ -10\le \gamma \le 10 $, the significance of $ ABS_\gamma $ is examined on the data set of octane isomers for predicting six selected physicochemical properties of the mentioned compounds; promising results are obtained when the approximated value of $ \gamma $ belongs to the set $ \{-3, 1, 3\} $. The effect of the addition of an edge between two non-adjacent vertices of a graph under $ ABS_\gamma $ is also investigated. Moreover, the graphs possessing the maximum value of $ ABS_{\gamma} $, with $ \gamma > 0 $, are characterized from the set of all connected graphs of a fixed order and a fixed (ⅰ) vertex connectivity not greater than a given number or (ⅱ) matching number.
- topological index,
- chemical graph theory,
- atom-bond sum-connectivity,
- general atom-bond sum-connectivity
Citation: Abeer M. Albalahi, Zhibin Du, Akbar Ali. On the general atom-bond sum-connectivity index[J]. AIMS Mathematics, 2023, 8(10): 23771-23785. doi: 10.3934/math.20231210

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Abstract

This paper is concerned with a generalization of the atom-bond sum-connectivity (ABS) index, devised recently in [A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081-2093]. For a connected graph $ G $ with an order greater than $ 2 $, the general atom-bond sum-connectivity index is represented as $ ABS_\gamma(G) $ and is defined as the sum of the quantities $ (1-2(d_x+d_y)^{-1})^{\gamma} $ over all edges $ xy $ of the graph $ G $, where $ d_x $ and $ d_y $ represent the degrees of the vertices $ x $ and $ y $ of $ G $, respectively, and $ \gamma $ is any real number. For $ -10\le \gamma \le 10 $, the significance of $ ABS_\gamma $ is examined on the data set of octane isomers for predicting six selected physicochemical properties of the mentioned compounds; promising results are obtained when the approximated value of $ \gamma $ belongs to the set $ \{-3, 1, 3\} $. The effect of the addition of an edge between two non-adjacent vertices of a graph under $ ABS_\gamma $ is also investigated. Moreover, the graphs possessing the maximum value of $ ABS_{\gamma} $, with $ \gamma > 0 $, are characterized from the set of all connected graphs of a fixed order and a fixed (ⅰ) vertex connectivity not greater than a given number or (ⅱ) matching number.

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