upper bounds for the reduced second zagreb index of graphs

The graph invariant rm2, known under the name reduced second zagreb index, is defined as rm2(g) = ∑ uv∈e(g)^(dg(u) − 1)(dg(v) − 1), where dg(v) is the degree of the vertex v of the graph g. in this paper, we give a tight upper bound of rm2 for the class of graphs of order n and size m with at least one dominating vertex. also, we obtain sharp upper bounds on rm2 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 rm2 for all k-apex trees of order n. moreover, the corresponding extremal graphs are characterized.