On the reduced second Zagreb index of graphs

For a connected graph G, the reduced second Zagreb index is defined asRM2=RM2(G)= ∑uv∈E(G)(dG(u)−1)(dG(v)−1),where dG(u) and dG(v) denote the degrees of vertices u and v in graph G, respectively. We determine the extremal graphs among all connected graphs of order n and with n2−m edges with respect to RM2, and obtain the first ten largest RM2 for connected graphs of order n. Moreover, we characterize the sharp upper bound on RM2 for all connected bipartite graphs with given connectivity k. Finally, the extremal graphs with respect to RM2 are determined in all connected bipartite graphs with matching number q.