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General sum-connectivity index of trees with given number of branching vertices | ||
| Transactions on Combinatorics | ||
| دوره 12، شماره 4، اسفند 2023، صفحه 227-238 اصل مقاله (447.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2022.133548.1987 | ||
| نویسنده | ||
| Tomas Vetrik* | ||
| Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa | ||
| چکیده | ||
| In 2015, Borovi'{c}anin presented trees with the smallest first Zagreb index among trees with given number of vertices and number of branching vertices. The first Zagreb index is obtained from the general sum-connectivity index if $a = 1$. For $a \in \mathbb{R}$, the general sum-connectivity index of a graph $G$ is defined as $\chi_{a} (G) = \sum_{uv\in E(G)} [d_G (u) + d_G (v)]^{a}$, where $E(G)$ is the edge set of $G$ and $d_G (v)$ is the degree of a vertex $v$ in $G$. We show that the result of Borovi'{c}anin cannot be generalized for the general sum-connectivity index ($\chi_{a}$ index) if $0 < a < 1$ or $a > 1$. Moreover, the sets of trees having the smallest $\chi_a$ index are not the same for $0 < a < 1$ and $a > 1$. Among trees with given number of vertices and number of branching vertices, we present all the trees with the smallest $\chi_a$ index for $0 < a < 1$ and $a > 1$. Since the hyper-Zagreb index is obtained from the $\chi_a$ index if $a = 2$, results on the hyper-Zagreb index are corollaries of our results on the $\chi_a$ index for $a > 1$. | ||
| کلیدواژه‌ها | ||
| degree؛ extremal graph؛ hyper-Zagreb index | ||
| مراجع | ||
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