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Solution to the minimum harmonic index of graphs with given minimum degree | ||
| Transactions on Combinatorics | ||
| مقاله 3، دوره 7، شماره 2، شهریور 2018، صفحه 25-33 اصل مقاله (232.56 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2017.101076.1462 | ||
| نویسندگان | ||
| Meili Liang؛ Bo Cheng؛ Jianxi Liu* | ||
| Guangdong University of Foreign Studies | ||
| چکیده | ||
| The harmonic index of a graph $G$ is defined as $ H(G)=\sum\limits_{uv\in E(G)}\frac{2}{d(u)+d(v)}$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. Let $\mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with minimum degree at least $k$. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among $\mathcal{G}(n,k)$. We solve the problem for each integer $k (1\le k\le n/2)$ and show the corresponding extremal graph is the complete split graph $K_{k,n-k}^*$. This result together with our previous result which solve the problem for each integer $k (n/2 \le k\le n-1)$ give a complete solution of the problem. | ||
| کلیدواژه‌ها | ||
| harmonic index؛ minimum degree؛ extremal graphs | ||
| مراجع | ||
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[9] L. Zhong, The harmonic index on unicyclic graphs, Ars Combin., 104 (2012) 261–269. | ||
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آمار تعداد مشاهده مقاله: 2,077 تعداد دریافت فایل اصل مقاله: 492 |
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