On the difference between geometric-arithmetic index and atom-bond connectivity index for trees

Volume 22, Issue 1, pp 49--58 http://dx.doi.org/10.22436/jmcs.022.01.05

Publication Date: June 29, 2020 Submission Date: April 16, 2020 Revision Date: April 28, 2020 Accteptance Date: May 16, 2020

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Authors

Wan Nor Nabila Nadia Wan Zuki - Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia. Roslan Hasni - Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia. Nor Hafizah Md. Husin - Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia. Zhibin Du - School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong, P. R. China. Abdul Raheem - Department of Higher Education, Govt. Postgraduate College Asghar Mall Rawalpindi, Pakistan.

Abstract

Let \(G\) be a simple and connected graph with vertex set \(V(G)\) and edge set \(E(G)\). The geometric-arithmetic index and atom-bond connectivity index of graph \(G\) are defined as \(GA(G)=\sum_{uv\in E(G)} \frac{2\sqrt{d_ud_v}}{d_u + d_v}\) and \(ABC(G)=\sum_{uv\in E(G)} \sqrt{\frac{d_u+d_v-2}{d_ud_v}}\), respectively, where the summation extends over all edges \(uv\) of \(G\), and \(d_u\) denotes the degree of vertex \(u\) in \(G\). Let \((GA-ABC)(G)\) denote the difference between \(GA\) and \(ABC\) indices of \(G\). In this note, we determine \(n\)-vertex binary trees with first three minimum \(GA-ABC\) values. We also present a lower bound for \(GA-ABC\) index of molecular trees with fixed number of pendant vertices.

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Keywords

• Atom-bond connectivity index
• geometric-arithmetic index
• trees

MSC

•  05C35
•  05C90

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