Multiplicity lists for symmetric matrices whose graphs have few missing edges

We characterize the possible lists of multiplicities occurring among the eigenvalues of real symmetric (or Hermitian) matrices whose graph is one of Kn, Kn less an edge, or both possibilities for Kn less two edges. The lists are quite different from those for trees. Some construction techniques are developed here and additional results with more missing edges are given, including the case of several independent edges.