Some Properties on the Harmonic Index of Molecular Trees

The harmonic index of a graph is defined as the sum of weights of all edges of , where denotes the degree of the vertex in . In this paper, some general properties of the harmonic index for molecular trees are explored. Moreover, the smallest and largest values of harmonic index for molecular trees with given pendent vertices are provided, respectively. 1. Introduction Let be a simple graph with vertex set and edge set . Its order is , denoted by . Let and be the degree and the set of neighbors of , respectively. The harmonic index of is defined in [1] as where the summation goes over all edges of . This index was extensively studied recently. For example, Zhong [2, 3] and Zhong and Xu [4] determined the minimum and maximum values of the harmonic index for simple connected graphs, trees, unicyclic graphs, and bicyclic graphs, respectively. Some upper and lower bounds on the harmonic index of a graph were obtained by Ilic [5]. Xu [6] and Deng et al. [7, 8] established some relationship between the harmonic index of a graph and its topological indices, such as Randi？ index, atom-bond connectivity index, chromatic number, and radius, respectively. Wu et al. [9] determined the graph with minimum harmonic index among all the graphs (or all triangle-free graphs) with minimum degree at least two. More information on the harmonic index of a graph can be found in [10]. The general sum-connectivity index of was proposed by Du et al. in [11] and defined as Clearly, . Du et al. [11] determined the maximum value and the corresponding extremal trees for the general sum-connectivity indices of trees for , where is the unique root of the equation . However, they did not consider the general sum-connectivity indices with . A molecular tree is a tree with maximum degree at most four. It models the skeleton of an acyclic molecule [12]. As far as we know, the mathematical properties of related indices for molecular trees have been studied extensively. For example, Gutman et al. [13, 14] determined the molecular trees with the first maximum, the second maximum, and the third maximum Randi？ indices, respectively. Du et al. [15] further determined the fourth maximum Randi？ index for molecular trees. Li et al. [16, 17] obtained the lower and upper bounds for the general Randi？ index for molecular trees and determined the molecular tree with minimum general Randi？ index among molecular trees with given pendant vertices. The graphs with maximum and minimum sum-connectivity indices among molecular trees with given pendant vertices were determined in Xing et al. [18]. In this