论文信息 - Change in vertex status after removal of another vertex in the general setting

Change in vertex status after removal of another vertex in the general setting

Abstract In the theory of multiplicities for eigenvalues of symmetric matrices whose graph is a tree, it proved very useful to understand the change in status (Parter, neutral, or downer) of one vertex upon removal of another vertex of given status (both in case the two vertices are adjacent or non-adjacent). As the subject has evolved toward the study of more general matrices, over more general fields, with more general graphs, it is appropriate to resolve the same type of question in the more general settings. “Multiplicity” now means geometric multiplicity. Here, we give a complete resolution in three more general settings and compare these with the classical case (216 “Yes” or “No” results). As a consequence, several unexpected insights are recorded.

Charles R. Johnson | Carlos M. Saiago | Kenji Toyonaga

[1] Charles R. Johnson,et al. ON THE MINIMUM NUMBER OF DISTINCT EIGENVALUES FOR A SYMMETRIC MATRIX WHOSE GRAPH IS A GIVEN TREE , 2002 .

[2] Charles R. Johnson,et al. The Parter-Wiener Theorem: Refinement and Generalization , 2003, SIAM J. Matrix Anal. Appl..

[3] Charles R. Johnson,et al. Eigenvalues, Multiplicities and Graphs , 2018 .

[4] Charles R. Johnson,et al. Hermitian Matrices, Eigenvalue Multiplicities, and Eigenvector Components , 2005, SIAM J. Matrix Anal. Appl..

[5] Charles R. Johnson,et al. On the relative position of multiple eigenvalues in the spectrum of an Hermitian matrix with a given graph , 2003 .

[7] Shaun M. Fallat,et al. ON TWO CONJECTURES REGARDING AN INVERSE EIGENVALUE PROBLEM FOR ACYCLIC SYMMETRIC MATRICES , 2004 .

[8] Charles R. Johnson,et al. Changes in vertex status and the fundamental decomposition of a tree relative to a multiple (parter) eigenvalue , 2017, Discret. Appl. Math..

[9] Carlos M. Saiago,et al. Diameter minimal trees , 2016 .

[10] Charles R. Johnson,et al. The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree , 2018, Linear Algebra and its Applications.

[11] Charles R. Johnson,et al. Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field , 2018 .

[12] Charles R. Johnson,et al. Geometric Parter–Wiener, etc. theory , 2018 .