Extremal Trees for the Geometric-Arithmetic Index with the Maximum Degree

Abstract

Abstract For a graph G, the geometric-arithmetic index of G, denoted by GA(G), is defined as the sum of the quantities 2 √ dx × dy/(dx + dy) over all edges xy ∈ E(G). Here, dx indicates the vertex degree of x. For every tree T of order n ≥ 3, Vukičević and Furtula [J. Math. Chem. 46 (2009) 1369–1376] demonstrated that GA(T ) ≤ 4 √ 2 3 + (n − 3). This result is extended in the present paper. Particularly, for any tree T of order n ≥ 5 and maximum degree ∆, it is proved that