On the Atom-Bond Connectivity Index of Two-Trees

DOI：10.3770/j.issn:2095-2651.2016.02.002

 作者 单位 于四勇 青海师范大学计算机学院, 青海 西宁 810008 赵海兴 青海师范大学数学系, 青海 西宁 810008 毛亚平 青海师范大学数学系, 青海 西宁 810008 肖玉芝 青海师范大学计算机学院, 青海 西宁 810008

设$G$是一个图,图G的ABC指标（atom-bond connectivity）由Estrada, Torres, Rodr\'{\i}guez 和 Gutman于1998提出，其定义如下:$$ABC(G)\sqrt{\frac{1}{d_i}+\frac{1}{d_j}-\frac{2}{{d_i}{d_j}}},$$和式取遍$G$所有的边 ${v_i}{v_j}$, $d_i$表示顶点$v_i$的度. 本文中给出了具有n个顶点的二树ABC指标的上界,即$ABC(G)\le(2n-4)\frac{\sqrt{2}}{2}+\frac{\sqrt{2n-4}}{n-1}$, 并确定了具有第一大和第二大ABC指标的二树.

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.