## Implicit construction of multiple eigenvalues for trees

Charles R. Johnson, Brian D. Sutton, and Andrew J. Witt
*Linear Multilinear Algebra* 57 (2009), no. 4, 409–420.

We are generally concerned with the possible lists of multiplicities
for the eigenvalues of a real symmetric matrix with a given graph.
Many restrictions are known, but it is often problematic to construct
a matrix with desired multiplicities, even if a matrix with such multiplicities exists. Here, we develop a technique for construction using
the implicit function theorem in a certain way. We show that the technique works for a large variety of trees, give examples and determine
all possible multiplicities for a large class of trees for which this was
not previously known.

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