Upper bounds for the reduced second zagreb index of graphs

The graph invariant RM_۲‎, ‎known under the name reduced second Zagreb index‎, ‎is defined as RM_۲(G)=\sum_{uv\in E(G)}(d_G(u)-۱)(d_G(v)-۱)‎, ‎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_۲ 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_۲ 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_۲ for all k-apex trees of order n‎. ‎Moreover‎, ‎the corresponding extremal graphs are characterized‎.

Keywords:

‎Reduced second Zagreb index‎ , ‎pendant vertex‎ , ‎dominating vertex‎ , ‎k-apex tree