On the relative position of multiple eigenvalues in the spectrum of an Hermitian matrix with a given graph

Charles R. Johnson, António Leal Duarte, Carlos M. Saiago, Brian D. Sutton, and Andrew J. Witt

Linear Algebra Appl. 363 (2003), 147–159.

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalues in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity, are investigated.