Let G be a nontrivial edge-colored connected graph. An edge-cut R of G is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph G is rainbow disconnected if for every two vertices u and v, there exists a u − v rainbow cut. For a connected graph G, the rainbow disconnection number of G, denoted by rd(G), is defined as the smallest number of colors that are needed in order to make G rainbow disconnected. In this paper, we first solve a conjecture that determines the maximum size of a connected graph G of order n with rd(G) = k for given integers k and n with 1 ≤ k ≤ n − 1, where n is odd, posed by Chartrand et al. in [5]. Secondly, we discuss bounds of the rainbow disconnection numbers for complete multipartite graphs, critical graphs, minimal graphs with respect to chromatic index and regular graphs, and give the rainbow disconnection numbers for several special graphs. Finally, we get the Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We prove that if G and G are both connected, then nFurthermore, examples are given to show that the upper bounds are sharp for n ≥ 6, and the lower bounds are sharp when G = G = P 4 .

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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 bicyclic conjugated molecular graphs (bicyclic graphs with perfect matching). Also a sharp lower bound on the harmonic index of bicyclic graphs is given in terms of the order and given size of matching.

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