Ordering Graphs by the Augmented Zagreb Indices

DOI：10.3770/j.issn:2095-2651.2015.02.001

 作者 单位 黄宇飞 广州民航职业技术学院数学教学部, 广东 广州 510403 柳柏濂 华南师范大学 数学科学学院, 广东 广州 510631

近年来, B. Furtula等学者提出了一个对辛烷和庚烷生成热的研究有较大价值的预测指标--图$G$的增强型萨格勒布指数($AZI$指数), 其定义为:$$AZI(G)=\sum_{uv\in E(G)}(\frac{d_ud_v}{d_u+d_v-2})^3,$$其中$E(G)$是图$G$的边集, $d_u$和$d_v$分别记边$uv\in E(G)$的两个端点$u$和$v$的度数. 在本文中, 我们分别得到了AZI指数前$5$大和前$2$小的$n$阶连通图; 而且, 我们分别确定了AZI 指数前$3$小的$n$阶树; AZI指数前$2$小的$n$阶单圈图; AZI指数最小的$n$阶双圈图.

Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index (AZI index) of a graph $G$, which is defined as $$\AZI(G)=\sum_{uv\in E(G)}\big(\frac{d_ud_v}{d_u+d_v-2}\big)^3,$$ where $E(G)$ is the edge set of $G$, $d_u$ and $d_v$ are the degrees of the terminal vertices $u$ and $v$ of edge $uv$, respectively. In this paper, we obtain the first five largest (resp., the first two smallest) AZI indices of connected graphs with $n$ vertices. Moreover, we determine the trees of order $n$ with the first three smallest AZI indices, the unicyclic graphs of order $n$ with the minimum, the second minimum AZI indices, and the bicyclic graphs of order $n$ with the minimum AZI index, respectively.