On $ ABC $ energy and its application to anticancer drugs

Alaa Altassan; Muhammad Imran; Bilal Ahmad Rather; Alaa Altassan; Muhammad Imran; Bilal Ahmad Rather

doi:10.3934/math.20231105

AIMS Mathematics

2023, Volume 8, Issue 9: 21668-21682. doi: 10.3934/math.20231105

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On $ ABC $ energy and its application to anticancer drugs

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain 15551, Abu Dhabi, United Arab Emirates

Received: 01 March 2023 Revised: 19 June 2023 Accepted: 25 June 2023 Published: 07 July 2023
MSC : 05C50, 05C92, 92E10, 15A18

For a simple connected graph $ \Gamma $ with node set $ V(\Gamma) = \{w_{1}, w_{2}, \dots, w_{n}\} $ and degree sequence $ d_{i} $, the atom-bond connectivity ($ ABC $) matrix of $ \Gamma $ has an $ (ij) $-th entry $ \sqrt{\frac{d_{i}+d_{j}-2}{d_{i}d_{j}}} $ if $ w_{i} $ is adjacent to $ w_{j} $ and $ 0 $, otherwise. The multiset of all eigenvalues of $ ABC $ matrix is known as the $ ABC $ spectrum and their absolute sum is known as the $ ABC $ energy of $ \Gamma. $ Two graphs of same order are known as $ ABC $ equienergetic if they have the same $ ABC $ energy but share different $ ABC $ spectrum. We describe the $ ABC $ spectrum of some special graph operations and as an application, we construct the $ ABC $ equienergetic graphs. Further, we give linear regression analysis of $ ABC $ index/energy with the physical properties of anticancer drugs. We observe that they are better correlated with $ ABC $-energy.
- adjacency matrix,
- $ ABC $ matrix,
- atom-bond connectivity,
- equienergetic graphs,
- correlation
Citation: Alaa Altassan, Muhammad Imran, Bilal Ahmad Rather. On $ ABC $ energy and its application to anticancer drugs[J]. AIMS Mathematics, 2023, 8(9): 21668-21682. doi: 10.3934/math.20231105

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Abstract

For a simple connected graph $ \Gamma $ with node set $ V(\Gamma) = \{w_{1}, w_{2}, \dots, w_{n}\} $ and degree sequence $ d_{i} $, the atom-bond connectivity ($ ABC $) matrix of $ \Gamma $ has an $ (ij) $-th entry $ \sqrt{\frac{d_{i}+d_{j}-2}{d_{i}d_{j}}} $ if $ w_{i} $ is adjacent to $ w_{j} $ and $ 0 $, otherwise. The multiset of all eigenvalues of $ ABC $ matrix is known as the $ ABC $ spectrum and their absolute sum is known as the $ ABC $ energy of $ \Gamma. $ Two graphs of same order are known as $ ABC $ equienergetic if they have the same $ ABC $ energy but share different $ ABC $ spectrum. We describe the $ ABC $ spectrum of some special graph operations and as an application, we construct the $ ABC $ equienergetic graphs. Further, we give linear regression analysis of $ ABC $ index/energy with the physical properties of anticancer drugs. We observe that they are better correlated with $ ABC $-energy.

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