On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree Original Research Article

We consider the general problem of determining which lists of multiplicities for the eigenvalues occur among Hermitian matrices the graph of whose off-diagonal entries is a given tree. Several restrictions are cited and a construction strategy is given. Together, these are sufficient to characterize all lists for each tree in two infinite classes: the double paths and generalized stars, and to tabulate all lists for trees on fewer than nine vertices. Such tables should be useful for formulating and dispelling general conjectures.