Minimum harmonie indices of trees and unicyclic graphs with given number of pendant vertices and diameter

Abstract

The harmonic index H(G) of a graph G is defined as the sum of weights of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we give sharp lower bounds for harmonic indices of trees and unicyclic graphs with n vertices and k pendant vertices, and characterize the corresponding extremal graphs. Furthermore, we also determine the smallest harmonic index of trees and unicyclic graphs with n vertices and diameter D(G).