The Second Geometric-Arithmetic Index For Trees and Unicyclic Graphs

Let $g$ be a finite and simple graph with edge set $e(g)$. the second geometricarithmetic index is defined as $ga_2(g)=sum_{uvin e(g)}frac{2sqrt{n_un_v}}{n_u+n_v}$, 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.