Ordered multiplicity lists for eigenvalues of symmetric matrices whose graph is a linear tree

We consider the class of trees for which all vertices of degree at least 3 lie on a single induced path of the tree. For such trees, a new superposition principle is proposed to generate all possible ordered multiplicity lists for the eigenvalues of symmetric (Hermitian) matrices whose graph is such a tree. It is shown that no multiplicity lists other than these can occur and that for two subclasses all such lists do occur. Important contrasts with trees outside the class are given, and it is shown that several prior conjectures about multiplicity lists, including the Degree Conjecture, follow from our superposition principle.

[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.  Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree , 2010, Discret. Appl. Math..

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

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

[5]  Gerry Wiener.  Spectral multiplicity and splitting results for a class of qualitative matrices , 1984 .

[6]  Charles R. Johnson,et al.  Implicit construction of multiple eigenvalues for trees , 2009 .

[7]  S. Parter.  On the Eigenvalues and Eigenvectors of a Class of Matrices , 1960 .

[8]  G. Golub,et al.  Structured inverse eigenvalue problems , 2002, Acta Numerica.

[9]  Charles R. Johnson,et al.  The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree , 1999 .

[10]  Charles R. Johnson,et al.  On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree , 2002 .

[11]  Charles R. Johnson,et al.  The implicit construction of multiplicity lists for classes of trees and verification of some conjectures , 2013 .

[12]  Charles R. Johnson,et al.  Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars , 2003 .

[13]  Shaun M. Fallat,et al.  On acyclic and unicyclic graphs whose minimum rank equals the diameter , 2008 .

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

[15]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.