三类不含拉普拉斯特征值1的树  

Three classes of trees without Laplacian eigenvalue 1

作　　者：王丽萍 李沐春[1] WANG Li-ping;LI Mu-chun(Institute of Applied Mathematics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学应用数学研究所,甘肃兰州730070

出　　处：《广州大学学报（自然科学版）》2023年第1期17-24,共8页Journal of Guangzhou University:Natural Science Edition

基　　金：国家自然科学基金资助项目(11961041);甘肃省自然科学基金资助项目(21JR11RA065)。

摘　　要：设A(G)为图G的邻接矩阵,D(G)为图G的度对角矩阵,称L(G)=D(G)-A(G)为图G的拉普拉斯矩阵,则特征多项式∅G(μ)=det(μI-L(G))的所有根称为图G的拉普拉斯特征值。一个端点的度不小于3,另一个端点的度等于1的路,被称为外部路。对于任意图G,如果G的外部路上包含P_(3)子图,则删除P_(3)不影响图G中拉普拉斯特征值1的重数。通过递归删除外部路上的P_(3),刻画了不含拉普拉斯特征值1的星型树、双星树和三星树。Let A(G)be the adjacent matrix of G,and D(G)be the diagonal matrix,the matrix L(G)=D(G)-A(G)is called the Laplacian matrix.Then all roots of the characteristic polynomial∅G(μ)=det(μI-L(G))are called Laplacian eigenvalues of G.A path with a vertex of degree not less than 3 and the other vertex of degree equal to 1 is called the external path of a graph.For any graph G,if the external path of G contains a P_(3) subgraph,then pruning off P_(3) s does not change the multiplicity of Laplacian eigenvalue 1 in graph G.In this paper,the starlike trees,the double-starlike trees and the three-starlike trees without a Laplacian eigenvalue 1 are completely characterized by deleting P_(3) recursively from external paths of those graphs.

关 键 词：树 拉普拉斯矩阵 特征值 重数 零度 

分 类 号：O157.5[理学—数学]