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Some remarks on the sum of powers of the degrees of graphs | ||
| Transactions on Combinatorics | ||
| دوره 10، شماره 1، خرداد 2021، صفحه 63-71 اصل مقاله (233.13 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2020.122877.1726 | ||
| نویسندگان | ||
| Emina I. Milovanovic؛ Marjan Matejic؛ Igor Z. Milovanovic* | ||
| Faculty of Electronic Engineering, University of Niˇ s, P.O.73, Niˇ s, Serbia | ||
| چکیده | ||
| Let $G=(V,E)$ be a simple graph with $n\ge 3$ vertices, $m$ edges and vertex degree sequence $\Delta=d_1 \ge d_2 \ge \cdots \ge d_n=\delta>0$. Denote by $S=\{1, 2,\ldots,n\}$ an index set and by $J=\{I=(r_1, r_2,\ldots,r_k), | ,1\le r_1<r_2<\cdots<r_k\le n\}$ a set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $d_{I}=d_{r_1}+d_{r_2}+\cdots+d_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots<r_k\le n-1$, the sum of $k$ arbitrary vertex degrees, where $\Delta_{I}=d_{1}+d_{2}+\cdots+d_{k}$ and $\delta_{I}=d_{n-k+1}+d_{n-k+2}+\cdots+d_{n}$. We consider the following graph invariant $S_{\alpha,k}(G)=\sum_{I\in J}d_I^{\alpha}$, where $\alpha$ is an arbitrary real number, and establish its bounds. A number of known bounds for various topological indices are obtained as special cases. | ||
| کلیدواژهها | ||
| Graphs؛ vertex degrees؛ graph invariants | ||
| مراجع | ||
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