The possible multiplicities of the eigenvalues of an hermitian matrix whose graph is a tree

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 self-contained 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.