On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree

It is shown that for any tree T the minimum number of distinct eigenvalues of an Hermitian matrix whose graph is T (diagonal entries free) is at least the number of vertices in a longest path of T. This is another step toward the general problem of characterizing the possible multiplicities for a given graph. Related observations are made and the result facilitates a table of multiplicities for trees on fewer than 8 vertices.