The Maximum Balaban Index (Sum-Balaban Index) of Unicyclic Graphs

DOI：10.3770/j.issn:2095-2651.2014.04.002

 作者 单位 尤利华 华南师范大学数学科学学院，广东 广州 510631 董欣 华南师范大学数学科学学院，广东 广州 510631

连通图$G$的Balaban指数定义为: $J(G)=\frac{|E(G)| }{\mu +1}\sum\limits_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u)D_{G}(v)}}} },$其和Balaban指数定义为: $SJ(G)=\frac{|E(G)| }{\mu 1}\sum\limits_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u) D_{G}(v)}}} },$这里 $D_{G}(u)=\sum\limits_{w\in V(G)}d_{G}(u,w),$ $\mu$ 是图$G$的基本圈的数目.本文刻画了$n$阶单圈图中具有最大Balaban指数的图与具有最大和Balaban指数的图.

The Balaban index of a connected graph $G$ is defined as $$J(G)=\frac{|E(G)| }{\mu +1}\sum_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u)D_{G}(v)}}} },$$ and the Sum-Balaban index is defined as $$SJ(G)=\frac{|E(G)| }{\mu 1}\sum_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u) D_{G}(v)}}} },$$ where $D_{G}(u)=\sum_{w\in V(G)}d_{G}(u,w),$ and $\mu$ is the cyclomatic number of $G$. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on $n$ vertices are characterized, respectively.