If \( G \) is a simple, connected  undirected graph and \( H(G) \) is the set of Hermitian matrices with graph \( G \), there has been long interest in the possible multiplicities of the eigenvalues of matrices in \( H(G) \), with initial work by Leal and myself starting the subject. In the case of trees, there is a dualc characterization of the maximum multiplicity (of interest to many). It is both the path cover number and may be calculated by deleting vertices to leave only paths, in such a way that the number of paths less the number of deleted vertices is a maximum. We discuss recent work that a) characterizes the second maximum multiplicity for trees in a parallel way and b) characterizes the maximum multiplicity for general graphs, also by deleting vertices.