Abstract
An important theorem about the existence of principal submatrices, of an Hermitian matrix whose graph is a tree, in which the multiplicity of an eigenvalue goes up was largely developed inseparate papers by Parter and Wiener. Here, the prior work isfully stated, then generalized with a selfcontained proof. The more complete result is then used to better understand the possible multiplicities of the eigenvalues of matrices whose graph is a tree. The relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvaluein the ordered spectrum are discussed in detail.
The Downer Branch mechanism, is a way to identify a (Parter) vertex of a tree $T$ (of a matrix $A$), whose removal from $T$ increases the multiplicity of an eigenvalue of $A$. Such vertices play an important role on the identification of the possible multiplicities of the eigenvalues of a matrix whose graphis a tree. We discuss the relation between multiple eigenvalues, Parter vertices and cardinality of Parter sets.
Special attention is given to a certain class of trees: the paths, generalized stars and double generalized stars. The possible multiplicities of the eigenvalues of a matrix whose graph is a path is well known, as each eigenvalue of such a matrix has multiplicity 1. However it was important to better understand the possible multiplicities of an eigenvalue of principal submatrices of a matrix whose graph is a path. We characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star or a double generalized star. Here, the inverse eigenvalue problem for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with aconjecture that determination of the possible ordered multiplicities is equivalent to the inverse eigenvalue problem for a given tree. Moreover, a key spectral feature of the inverse eigenvalue problem in the case of generalized stars is shown to characterize them among trees.
The Downer Branch mechanism, is a way to identify a (Parter) vertex of a tree $T$ (of a matrix $A$), whose removal from $T$ increases the multiplicity of an eigenvalue of $A$. Such vertices play an important role on the identification of the possible multiplicities of the eigenvalues of a matrix whose graphis a tree. We discuss the relation between multiple eigenvalues, Parter vertices and cardinality of Parter sets.
Special attention is given to a certain class of trees: the paths, generalized stars and double generalized stars. The possible multiplicities of the eigenvalues of a matrix whose graph is a path is well known, as each eigenvalue of such a matrix has multiplicity 1. However it was important to better understand the possible multiplicities of an eigenvalue of principal submatrices of a matrix whose graph is a path. We characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star or a double generalized star. Here, the inverse eigenvalue problem for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with aconjecture that determination of the possible ordered multiplicities is equivalent to the inverse eigenvalue problem for a given tree. Moreover, a key spectral feature of the inverse eigenvalue problem in the case of generalized stars is shown to characterize them among trees.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  4 Mar 2004 
Place of Publication  Lisboa 
Publisher  
Publication status  Published  2004 