  Harary and Schwenk posed the problem forty years ago: Which graphs have
distinct adjacency eigenvalues? In this paper, we obtain a necessary and
sufficient condition for an Hermitian matrix with simple spectral radius and
distinct eigenvalues. As its application, we give an algebraic characterization
to the Harary-Schwenk's problem. As an extension of their problem, we also
obtain a necessary and sufficient condition for a positive semidefinite matrix
with simple least eigenvalue and distinct eigenvalues, which can provide an
algebraic characterization to their problem with respect to the (normalized)
Laplacian matrix.

Combinatorics