University of Isfahan Transactions on Combinatorics 2251-8657 10 3 2021 09 01 Upper bounds for the reduced second zagreb index of graphs 137 148 25174 10.22108/toc.2020.125478.1774 EN Batmend Horoldagva Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia 0000-0003-3417-2612 Tsend-Ayush Selenge Department of Mathematics, National University of Mongolia, P.O.Box 187/46A, Ulaanbaatar, Mongolia Lkhagva Buyantogtokh Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia Shiikhar Dorjsembe Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia Journal Article 2020 10 18 The graph invariant $RM_2$‎, ‎known under the name reduced second Zagreb index‎, ‎is defined as $RM_2(G)=\sum_{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)$‎, ‎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. https://toc.ui.ac.ir/article_25174_4be757e0eb219336680ac34256e5727e.pdf