论文标题
$a_α$频谱半径，具有给定的独立号$ n-4 $
The $A_α$ spectral radius with given independence number $n-4$
论文作者
论文摘要
令$ g $为具有邻接矩阵$ a（g）$和学位对角线矩阵$ d（g）$的图表。 2017年，Nikiforov [应用。肛门。离散数学，11（2017）81--107]定义了矩阵$a_α（g）=αd（g） +（1-α）a（g）a（g）$，用于[0,1] $中的任何实际$α\。 $ a（g）$的最大特征值称为$ g $的频谱半径，而最大的$a_α（g）$的特征值称为$a_α$ spectral speptral半径$ g $。令$ \ Mathcal {g} _ {n，i} $是订单$ n $带有独立数字$ i $的图表。最近，对于$ \ Mathcal {g} _ {n，i} $具有最小值或最大$ a $，$ q $和$a_α$ peptral半径的所有图 $i\in\{1,2,\lfloor\frac{n}{2}\rfloor\,\lceil\frac{n}{2}\rceil+1,n-3,n-2,n-1\}$, there are some results have been given by Xu, Li and Sun et al., respectively.在2021年，Luo和Guo [离散数学，345（2022）112778]确定了$ \ Mathcal {g} _ {n，n-4} $具有最小光谱半径的所有图。在本文中，我们在$ \ MATHCAL {G} _ {n，n-4} $中表征图形，分别为$α\ in [\ frac {1} {2} {2}，1）$的最小值和最大$a_α$ spectral radius。
Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_α(G) = αD(G) + (1-α)A(G)$ for any real $α\in[0,1]$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$, while the largest eigenvalue of $A_α(G)$ is called the $A_α$ spectral radius of $G$. Let $\mathcal{G}_{n,i}$ be the set of graphs of order $n$ with independence number $i$. Recently, for all graphs in $\mathcal{G}_{n,i}$ having the minimum or the maximum $A$, $Q$ and $A_α$ spectral radius where $i\in\{1,2,\lfloor\frac{n}{2}\rfloor\,\lceil\frac{n}{2}\rceil+1,n-3,n-2,n-1\}$, there are some results have been given by Xu, Li and Sun et al., respectively. In 2021, Luo and Guo [Discrete Math., 345 (2022) 112778] determined all graphs in $\mathcal{G}_{n,n-4}$ having the minimum spectral radius. In this paper, we characterize the graphs in $\mathcal{G}_{n,n-4}$ having the minimum and the maximum $A_α$ spectral radius for $α\in[\frac{1}{2},1)$, respectively.
