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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