The harmonic index of a graph G is defined as the sum of the weights 2d(u)+d(v) ofall edges uv of G, where d(u) denotes the degree of the vertex u in G. A graph G is calledquasi-tree, if there exists u ∈ V(G) such that G?u is a tree. The graphs called two-trees aredefined by recursion. The smallest two-tree is the complete graph on two vertices. A two-treeon n+1 vertices (where n 2) is obtained by adding a new vertex adjacent to the two endvertices of one edge in a two-tree on n vertices. In this work, the sharp lower and upper boundson the harmonic index of quasi-tree graphs are presented. Furthermore, the lower bound on theharmonic index of two-trees is presented, and the two-trees with the minimum and the secondminimum harmonic index, respectively, are determined.
展开▼
