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Upper bounds for the reduced second zagreb index of graphs | ||
| Transactions on Combinatorics | ||
| دوره 10، شماره 3، آذر 2021، صفحه 137-148 اصل مقاله (250.07 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2020.125478.1774 | ||
| نویسندگان | ||
| Batmend Horoldagva1؛ Tsend-Ayush Selenge* 2؛ Lkhagva Buyantogtokh1؛ Shiikhar Dorjsembe1 | ||
| 1Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia | ||
| 2Department of Mathematics, National University of Mongolia, P.O.Box 187/46A, Ulaanbaatar, Mongolia | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Reduced second Zagreb index؛ pendant vertex؛ dominating vertex؛ $k$-apex tree | ||
| مراجع | ||
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