The Second Geometric-arithmetic Index for Trees and Unicyclic Graphs

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran

2 Department of Mathematics, Gareziaeddin Center, Khoy Branch, Islamic Azad University, Khoy, Iran

3 Department of Mathematics, University of West Georgia, Carrollton GA 30082

Abstract

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=\sum_{uv\in E(G)}\frac{2\sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree of the tree. We also find a sharp upper bound for $GA_2(G)$, where $G$ is a unicyclic graph, in terms of the order, maximum degree and girth of $G$. In addition, we characterize the trees and unicyclic graphs which achieve the upper bounds.

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References

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Iranian Journal of Mathematical Chemistry
Volume 9, Issue 4
December 2018
Pages 279-287

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