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 geometric-arithmetic Index is defined as GA(2)(G) = Sigma(uv is an element of E(G)) 2 root n(u)n(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. (C) 2018 University of Kashan Press. All rights reserved