关于树补图的A_(α)-谱半径的一些极值结论  

Some extremal results on the A_(α)-spectral radius of the complement graphs of trees

作　　者：彭家荣 朱艳丽[1] 张蓝 PENG Jia-Rong;ZHU Yan-Li;ZHANG Lan(College of Math.and Inform.,South China Agricultural Univ.,Guangzhou 510642,China)

机构地区：[1]华南农业大学数学与信息学院,广东广州510642

出　　处：《高校应用数学学报（A辑）》2024年第4期493-500,共8页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(12271182);广东省自然科学基金(2022A1515011786,2024A1515011899);2022年国家级大学生创新训练项目(202210564068)。

摘　　要：设A(G)和D(G)分别表示图G的邻接矩阵和度对角矩阵,称A_(α)(G)=αD(G)+(1-α)A(G)为图G的A_(α)-矩阵,并称A_(α)(G)的最大特征值为图G的A_(α)-谱半径,其中α∈[0,1).图G的A_(α)-矩阵是图G的邻接矩阵和无符号Laplacian矩阵的共同推广.该文研究了树的补图中谱半径的排序问题,分别确定了最大度为△的n阶树的补图中A_(α)-谱半径的唯一极大和唯一极小图,还确定了n阶树的补图中唯一的A_(α)-谱半径极小图.在此基础上,得到了n阶树的补图中邻接谱半径的标尺定理(The Scalar Theorem).Let A(G)and D(G)denote the adjacency matrix and the diagonal matrix of the degrees of a simple graph G,respectively.For α∈[0,1),let A_(α)(G)=αD(G)+(1-α)A(G)be the Aα-matrix of the graph G,and the largest eigenvalue of Aα-matrix is called the A_(α)-spectral radius of G.The A_(α)-matrix of a graph G is a unified definition of the adjacency matrix and the signless Laplacian matrix of G.In this paper,the unique maximal and unique minimal extremal graph in the class of the complement graphs of trees with n vertices and maximum degree ∆ is determined,respectively.Consequently,the unique minimal graph in the class of the complement graphs of trees with n vertices is also determined.The Scalar Theorem of spectral radius in the class of the complement graphs of trees with n vertices is proved.

关 键 词：A_(α)-矩阵 谱半径 树 补图 标尺定理(The Scalar Theorem) 

分 类 号：O157.5[理学—数学]