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The harmonic index H(G) of a graph G is the sum of 2 d(u) + d(υ) over all edges uυ of G, where d(u) denotes the degree of a vertex u in G. In this paper, we give the minimum value of H(G) for graphs G with given minimum degree δ(G) ≥ 2 and characterize the corresponding extremal graph. Furthermore, we prove a best-possible lower bound on the harmonic index… (More)
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.
We say that a simple graph G is fractional independent-set-deletable k-factor-critical, shortly, fractional ID-k-factor-critical, if G − I has a fractional k-factor for every independent set I of G. Some sufficient conditions for a graph to be fractional ID-k-factor-critical are studied in this paper. Furthermore, we show that the result is best possible in… (More)
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)
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 first present a sharp lower bound on the harmonic index of unicyclic conjugated graphs (unicyclic graphs with a perfect matching). Also a sharp lower bound on the harmonic index of… (More)