An important theorem about the existence of principal submatrices of a Hermitian matrix whose graph is a tree, in which the multiplicity of an eigenvalue increases, was largely developed in separate papers by Parter and Wiener. Here, the prior work is fully stated, then generalized with a self-contained proof. The more complete result is then used to better understand the eigenvalue possibilities of reducible principal submatrices of Hermitian tridiagonal matrices. Sets of vertices, for which the multiplicity increases, are also studied.

We characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star (a tree in which at most one vertex has degree greater than 2) 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 a conjecture 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.

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The maximum multiplicity among eigenvalues of matrices with a given graph cannot generally be expressed in terms of the degrees of the vertices (even when the graph is a tree). Given are best possible lower and upper bounds, and characterization of the cases of equality in these bounds. A by-product is a sequential algorithm to calculate the exact maximum multiplicity by simple counting.

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