| In order to study the structure characteristics of a graph,many topological indices related to a graph are proposed,studying the relationship between the structural properties of a graph and its topological index is one of the popular topics in extremal graph theory.In this thesis,we consider one of popular topological indices—harmonic index of a graph G,which is defined as (?),where the summation goes over all edges uv of G.We study the extremal problem for the harmonic index of trees with a given number of segments and trees with a given number of branching vertices,respectively.The maximum and minimum values for the harmonic index of these two kinds of trees are determined,and the corresponding extremal graphs that reach those values are also constructed,respectively. |
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