Through a succession of results, it is known that if the graph of an Hermitian matrix A is a tree and if for some index j, λ ∈ σ (A) ∩ σ (A (j)), then there is an index i such that the multiplicity of λ in σ (A (i)) is one more than that in A. We exhibit a converse to this result by showing that it is generally true only for trees. In particular, it is shown that the minimum rank of a positive semidefinite matrix with a given graph G is ≤ n - 2 when G is not a tree. This raises the question of how the minimum rank of a positive semidefinite matrix depends upon the graph in general. © 2006 Elsevier B.V. All rights reserved.
Johnson, C. R., & Duarte, A. L. (2006). Converse to the Parter-Wiener theorem: The case of non-trees. Discrete Mathematics, 306(23 SPEC. ISS.), 3125–3129. https://doi.org/10.1016/j.disc.2005.04.025