Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

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 (IEP) 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 IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.