Upper bounds for the reduced second zagreb index of graphs
نویسندگان
1 Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia
2 Department of Mathematics, National University of Mongolia, P.O.Box 187/46A, Ulaanbaatar, Mongolia
3 Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia
4 Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia
doi
10.22108/toc.2020.125478.1774چکیده
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.
