On the harmonic index of bicyclic graphs

The harmonic index of a graph G, denoted by H(G), is defined as the sum of weights 2/[d(u) + d(v)] over all edges uv of G, where d(u) denotes the degree of a vertex u. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. 12 (2013) 716–726] proved that for any bicyclic graph G of order n ≥ 4, H(G) ≤ ⁿ − ¹ and 2 15 characterized all extremal bicyclic graphs. In this paper, we prove that for any bicyclic graph G of order n ≥ 4 and maximum degree ∆, and characterize all extremal bicyclic graphs.