University of KashanIranian Journal of Mathematical Chemistry2228-64899420181201The second geometric-arithmetic index for trees and unicyclic graphs2792878154410.22052/ijmc.2017.81079.1277ENN. DehgardiDepartment of Mathematics and Computer Science, Sirjan University of Technology,
Sirjan, IranH. AramDepartment of Mathematics,
Gareziaeddin Center, Khoy Branch, Islamic Azad University, Khoy, IranA. KhodkarDepartment of Mathematics, University of West Georgia, Carrollton GA 30082Journal Article20170401Let $G$ be a finite and simple graph with edge set $E(G)$. The<br /> second geometric-arithmetic index is defined as<br /> $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$,<br /> where $n_u$ denotes the number of vertices in $G$ lying closer to<br /> $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms<br /> of the order and maximum degree of the tree. We also find a sharp upper bound for $GA_2(G)$, where $G$<br /> is a unicyclic graph, in terms of the order, maximum degree and girth of $G$.<br /> In addition, we characterize the trees and unicyclic graphs which achieve the upper bounds.http://ijmc.kashanu.ac.ir/article_81544_d6ee54879d3b9af783c9e4a0e8b112b9.pdf