The harmonic index is one of the most important indices in chemical and mathematical fields. It's a variant of the Randic index which is the most successful molecular descriptor in structure-property and structure-activity relationships studies. The harmonic index gives somewhat better correlations with physical and chemical properties comparing with the well known Randic index. The harmonic index H(G) of a graph G is defined as the sum of the weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the unicyclic and bicyclic graphs with minimum and maximum harmonic index, and also characterize the corresponding extremal graphs. The unicyclic and bicyclic graphs with minimum harmonic index are S_n~+, S_n~1 respectively, and the unicyclic and bicyclic graphs with maximum are C_n, B_n or B_n' respectively. As a simple result, we present a short proof of one theorem in Applied Mathematics Letters 25 (2012) 561-566, that the trees with maximum and minimum harmonic index are the path P_n and the star S_n, respectively. Moreover, we give a further discussion about the property of the graphs with the maximum harmonic index, and show that the regular or almost regular graphs have the maximum harmonic index in connected graphs with n vertices and m edges.
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