给定悬挂点的非平衡符号图的最小拉普拉斯特征值  

THE LEAST LAPLACIAN EIGENVALUE OF UNBALANCED SIGNED GRAPHS OF FIXED ORDER AND GIVEN NUMBER OF PENDANT VERTICES

作　　者：汪赛[1,2] 王登银 田凤雷 WANG Sai;WONG Dein;TIAN Feng-lei(Xuhai College,China University of Mining and Technology,Xuzhou 221116,China;School of Mathematics,China University of Mining and Technology,Xuzhou 221116,China;School of Management,Qufu Normal University,Rizhao 276826,China)

机构地区：[1]中国矿业大学徐海学院,江苏徐州221116 [2]中国矿业大学数学学院,江苏徐州221116 [3]曲阜师范大学管理学院,山东日照276826

出　　处：《数学杂志》2020年第6期643-652,共10页Journal of Mathematics

基　　金：Supported by the National Natural Science Foundation of China(11971474);Natural Science Foundation of Shandong Province(ZR2019BA016).

摘　　要：符号图是边赋值为±1的一类图.设符号图Γ的拉普拉斯矩阵为L(Γ)=D(G)-A(Γ),这里D(G)表示度矩阵, A(Γ)表示符号图的邻接矩阵.Γ是平衡的当且仅当最小拉普拉斯特征值λn=0.因此当Γ非平衡时λn>0.本文研究了非平衡符号图的最小拉普拉斯特征值问题.利用图特征值的嫁接方法,获得了给定悬挂点非平衡符号图的最小拉普拉斯特征值,并且刻画了达到最小特征值的极图.Signed graphs are graphs whose edges get signs ±1 and, as for unsigned graphs,they can be studied by means of graph matrices. For a signed graph Γ we consider the Laplacian matrix defined as L(Γ) = D(G)-A(Γ), where D(G) is the matrix of vertex degrees of G and A(Γ)is the signed adjacency matrix. It is well known that a connected graph Γ is balanced if and only if the least Laplacian eigenvalues λn=0. Therefore, if a connected graph Γ is not balanced, then λn> 0. In this paper, we investigate how the least eigenvalue of the Laplacian of a signed graph changes by relocating a tree branch from one vertex to another. As an application, we determine the graph whose least laplacian eigenvalue attains the minimum among all connected unbalanced signed graphs of fixed order and given number of pendant vertices.

关 键 词：符号图 拉普拉斯 最小特征值 

分 类 号：O157.5[理学—数学]