Converse to the Parter–Wiener theorem: The case of non-trees

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))λ∈σ(A)∩σ(A(j)), then there is an index i   such that the multiplicity of λλ in σ(A(i))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⩽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.