| On the Atom-Bond Connectivity Index of Two-Trees |
| Received:February 09, 2015 Revised:September 14, 2015 |
| Key Words:
graph two-trees atom-bond connectivity index
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.\,61164005; 61440005; 11161037), the Program for Changjiang Scholars and Innovative Research Team in Universities (Grant No.\,IRT-15R40), Key Laboratory of Tibetan Information Processing of Ministry of Education of China, the Research Fund for the Chunhui Program of Ministry of Education of China (Grant No.\,Z2014022) and the Nature Science Foundation of Qinghai Province (Grant Nos.\,2013-Z-Y17; 2014-ZJ-907; 2014-ZJ-721). |
| Author Name | Affiliation | | Siyong YU | School of Computer, Qinghai Normal University, Qinghai 810008, P. R. China | | Haixing ZHAO | Department of Mathematics, Qinghai Normal University, Qinghai 810008, P. R. China | | Yaping MAO | Department of Mathematics, Qinghai Normal University, Qinghai 810008, P. R. China | | Yuzhi XIAO | School of Computer, Qinghai Normal University, Qinghai 810008, P. R. China |
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| Abstract: |
| The atom-bond connectivity $(ABC)$ index of a graph $G$, introduced by Estrada, Torres, Rodr\'{\i}guez and Gutman in 1998, is defined as the sum of the weights $\sqrt{\frac{1}{d_i}+\frac{1}{d_j}-\frac{2}{{d_i}{d_j}}}$ of all edges ${v_i}{v_j}$ of $G$, where $d_i$ denotes the degree of the vertex $v_i$ in $G$. In this paper, we give an upper bound of the $ABC$ index of a two-tree $G$ with $n$ vertices, that is, $ABC(G)\le(2n-4)\frac{\sqrt{2}}{2}+\frac{\sqrt{2n-4}}{n-1}$. We also determine the two-trees with the maximum and the second maximum ABC index. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2016.02.002 |
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