Title of article :
Upper bounds for the reduced second zagreb index of graphs
Author/Authors :
Horoldagva, Batmend Department of Mathematics - Mongolian National University of Education - Ulaanbaatar, Mongolia , Selenge, Tsend-Ayush Department of Mathematics - National University of Mongolia, Ulaanbaatar, Mongolia , Buyantogtokh, Lkhagva Department of Mathematics - Mongolian National University of Education - Ulaanbaatar, Mongolia , Dorjsembe, Shiikhar Department of Mathematics - Mongolian National University of Education - Ulaanbaatar, Mongolia
Abstract :
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.
Keywords :
Reduced second Zagreb index , pendant vertex , dominating vertex , k-apex tree
Journal title :
Transactions on Combinatorics
