Trees with Given Diameter and Minimum Second Geometric–Arithmetic Index∗

The second geometric-arithmetic index GA2(G) of a graph G was introduced recently by Fath-Tabar et al. [2] and is defined to be ∑ uv∈E(G) √ nu(e,G)nv(e,G) 1 2 [nu(e,G)+nv(e,G)] , where e = uv is one edge in G, and nu(e,G) denotes the number of vertices in G lying closer to u than to v. In this paper, we characterize the tree with the minimum GA2 index among the set of trees with given order and diameter. As applications, we deduce the trees with the minimum and second-minimum GA2 index among the set of trees of given order, respectively. In addition, all the trees minimizing the GA2 index have been shown to have minimum Szeged index and Wiener index, which deduced a result of [7] concerning the Wiener index of trees with given diameter.