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运筹学学报 ›› 2024, Vol. 28 ›› Issue (1): 121-130.doi: 10.15960/j.cnki.issn.1007-6093.2024.01.010

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关于赋权非正则图的Aα特征值和特征向量

何常香1, 王文燕1, 刘乐乐1,*()   

  1. 1. 上海理工大学理学院, 上海 200093
  • 收稿日期:2020-09-02 出版日期:2024-03-15 发布日期:2024-03-15
  • 通讯作者: 刘乐乐 E-mail:leliu@usst.edu.cn
  • 基金资助:
    上海市自然科学基金(12ZR1420300);国家自然科学基金(11101284);国家自然科学基金(11201303);国家自然科学基金(12001370)

On the eigenvalues and eigenvectors of Aα in weighted non-regular graphs

Changxiang HE1, Wenyan WANG1, Lele LIU1,*()   

  1. 1. College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
  • Received:2020-09-02 Online:2024-03-15 Published:2024-03-15
  • Contact: Lele LIU E-mail:leliu@usst.edu.cn

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摘要：

设$G_\omega=(G, \omega)$是一个赋权图, 其邻接矩阵和赋权度对角矩阵分别$A(G_\omega)$和$D(G_\omega)$。对于$\alpha\in[0, 1]$, $G_\omega$的$A_\alpha$-矩阵为$ A_\alpha(G_\omega)=\alpha D(G_\omega)+(1-\alpha)A(G_\omega)$。对于连通赋权非正则图$G_\omega$, 给出了其关于$A_\alpha$-特征值的一些界, 并得到了$A_\alpha$-谱半径对应的特征向量中最大分量与最小分量比值的下界。

关键词: 赋权图, Aα-矩阵, Aα-谱半径

Abstract:

Let $G_\omega=(G, \omega)$ be a weighted graph, whose adjacency matrix and weighted degree diagnoal matrix are $A(G_\omega)$ and $D(G_\omega)$, respectively. For given $\alpha \in [0, 1]$, the matrix $A_\alpha(G_\omega)=\alpha D(G_\omega)+(1-\alpha)A(G_\omega)$ is the $A_\alpha$- matrix of $G_\omega$. In this paper, we give some bounds on the $A_\alpha$-eigenvalue of connected weighted non-regular graphs $G_\omega$, and obtain the lower bound of the ratio of the largest component to the smallest component in the eigenvector of the $A_\alpha$- spectral radius.

Key words: weighted graph, Aα-matrix, Aα-spectral radius

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