Some results on the difference of the zagreb indices of a graph

The classical first and second Zagreb indices of a graph G are defined as M1(G)=∑v∈V(G)d(v)²and M2(G)=∑e=uv∈E(G)d(u)d(v); where d(v) is the degree of the vertex v of G: Recently, Furtula et al. ['On difference of Zagreb indices', Discrete Appl. Math. 178 (2014), 83-88] studied the difference of M1 and M2; and showed that this difference is closely related to the vertex-degree-based invariant RM2(G) =∑e=uv7isin;E(G)[d(u) . 1][d(v)-1], the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.