Upper bounds for the reduced second zagreb index of graphs

The graph invariant $RM_2$‎, ‎known under the name reduced second Zagreb index‎, ‎is defined as $RM_2(G)=sum_{uvin E(G)}(d_G(u)-1)(d_G(v)-1)$‎, ‎where $d_G(v)$ is the degree of the vertex $v$ of the graph $G$‎. ‎In this paper‎, ‎we give a tight upper bound of $RM_2$ for the class of graphs of order $n$ and size $m$ with at least one dominating vertex‎. ‎Also‎, ‎we obtain sharp upper bounds on $RM_2$ for all graphs of order $n$ with $k$ dominating vertices and for all graphs of order $n$ with $k$ pendant vertices‎. ‎Finally‎, ‎we give a sharp upper bound on $RM_2$ for all $k$-apex trees of order $n$‎. ‎Moreover‎, ‎the corresponding extremal graphs are characterized‎.