单圈图Sigma指数的极大值

On the Maximum Sigma Index of Unicyclic Graphs

摘要: 无向连通图$G$ 的Sigma指数定义为$\sigma \left(G\right)={\displaystyle {\sum }_{uv\in E\left(G\right)}{\left(d_G\left(u\right)-d_G\left(v\right)\right)}^2}$ ，其中，$d_G\left(u\right)$ 和$d_G\left(v\right)$ 分别表示顶点$u$ 和$v$ 在图$G$ 中的度。本文研究了$n$ 阶单圈图的Sigma指数，用不同的方法讨论了其最大值。在本文中，我们也得到了单圈图中Sigma指数第二大值，并刻画了其极图。

Abstract: The sigma of an undirected graph $G$ is defined as $\sigma \left(G\right)={\displaystyle {\sum }_{uv\in E\left(G\right)}{\left(d_G\left(u\right)-d_G\left(v\right)\right)}^2}$ , where $d_G\left(u\right)$ and $d_G\left(v\right)$ denote the degrees of vertices $u$ and $v$ , respectively. In this paper, we investigate the Sigma index of n-vertex unicyclic graphs and discuss its maximum value using different methods. Furthermore, we determine the second largest Sigma index and extremal graph among unicyclic graphs.