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.

Keywords

Main Subjects


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