On the harmonic index of the unicyclic and bicyclic graphs

The harmonic index is one of the most important indices in chemical and mathematical fields. It’s a variant of the Randić index which is the most successful molecular descriptor in structure-property and structureactivity relationships studies. The harmonic index gives somewhat better correlations with physical and chemical properties comparing with the well known Randić index. The harmonic indexH(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 1 n respectively, and the unicyclic and bicyclic graphs with maximum are Cn, Bn 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 Pn and the star Sn, 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 andm edges. Key–Words: The harmonic index; minimum; maximum; unicyclic graphs; bicyclic graphs