The harmonic index H(G) of a graph G is defined as the sum of 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 have determined the minimum and maximum harmonic indices of bicyclic graphs and characterized the corresponding graphs at which the ex-tremal harmonic indices are attained.
Let S be a finite set of positive integers. A mixed hypergraph H is a one-realization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. The smallest one-realization of a given set II, Discrete Math. 312 (2012) 2946–2951], we determined the minimum number of vertices of a 3-uniform bi-hypergraph which is a… (More)