The harmonic index H G of a graph G is defined as the sum of the weights 2 u v d d of all edges uv ofG , where u d denotes the degree of a vertex u inG . In this paper, we obtained some new relationships between harmonic index and first geometric-arithmetic index, sum connectivity index that this indices are important than another topological index. In addition, we determine the lower and upper bond for molecular graphs and unicyclic molecular graph. Also we give a characterization of the minimum harmonic index of graphs with maximum degree .
