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 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 inverse eigenvalue problem for a given tree. Moreover, a key spectral feature of the inverse eigenvalue problem in the case of generalized stars is shown to characterize them among trees. ∗Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA ([email protected]). †Dep. de Matemática, Univ. de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal ([email protected]). This research was supported by Centro de Matemática da Universidade de Coimbra. ‡Dep. de Matemática, Fac. de Ciências e Tecnologia da Univ. Nova de Lisboa, 2829-516 Quinta da Torre, Portugal ([email protected]). Research supported in part by Fundação para a Ciência e a Tecnologia, Portugal, through the research grant SFRH/BD/899/2000. Part of the research was done while visiting the College of William and Mary.