For the problem of understanding what multiplicities are possible for eigenvalues among real symmetric matrices with a given graph, constructing matrices with conjectured multiplicities is generally more difficult than finding constraining conditions. Here, the implicit function theorem method for constructing matrices with a given graph and given multiplicity list is refined and extended. In particular, the breadth of known circumstances in which the Jacobian is nonsingular is increased. This allows characterization of all multiplicity lists for binary, diametric, depth one trees. In addition the degree conjecture and a conjecture about the minimum number of multiplicities equal to 1 is proven for diametric trees. Finally, an intriguing conjecture about the eigenvalues of a matrix whose graph is a path and its submatrices is given, along with a discussion of some ides that would support a proof of the degree conjecture and the minimum number of 1\u27s conjecture, in general. (c) 2012 Elsevier Inc. All rights reserved
