University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 9 4 2018 12 01 The Second Geometric-arithmetic Index for Trees and Unicyclic Graphs 279 287 81544 10.22052/ijmc.2017.81079.1277 EN N. Dehgardi Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran H. Aram Department of Mathematics, Gareziaeddin Center, Khoy Branch, Islamic Azad University, Khoy, Iran A. Khodkar Department of Mathematics, University of West Georgia, Carrollton GA 30082 Journal Article 2017 04 01 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. https://ijmc.kashanu.ac.ir/article_81544_d6ee54879d3b9af783c9e4a0e8b112b9.pdf