# 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

### References

1 K. Ch. Das, I. Gutman and B. Furtula, On second geometric-arithmetic index of graphs, Iranian J. Math. Chem. 1 (2010), 17–28.
2 K. Ch. Das, I. Gutman and B. Furtula, On the first geometric-arithmetic index of graphs, Discrete Appl. Math. 159 (2011), 2030–2037.
3 H. Hua, Trees with given diameter and minimum second geometric-arithmetic index, J. Math. Chem. 64 (2010), 631–638.
4 A. Madanshekaf and M. Moradi, The first geometric-arithmetic index of some nanostar dendrimers, Iranian J. Math. Chem. 5 (2014), 1–6.
5 G. H. Fath-Tabar, B. Furtula and I. Gutman, A new geometric-arithmetic index, J. Math. Chem. 47 (2010), 477–486.
6 D. Vukicevic and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 4 (2009), 1369–1376.
7 Y. Yuan, B. Zhou and N. Trinajstic, On geometric-arithmetic index, J. Math. Chem. 47 (2010), 833–841.