Converse to the Parter--Wiener theorem: the case of non-trees (Q856860)

Let \(A(i)\) denote the matrix obtained by deleting the \(i\)th row and column of the matrix \(A\), and let \(m_A(\lambda )\) be the multiplicity of the eigenvalue \(\lambda\) of \(A\). The Parter--Wiener theorem, as extended by the authors and \textit{C.\ M.\ Saiago} [SIAM J.\ Matrix Anal.\ Appl.\ 25, 352--361 (2003; Zbl 1067.15003)], asserts that if the graph of a Hermitian matrix \(A\) is a tree and, for some index \(j\), \(\lambda \in \sigma (A)\cap \sigma (A(j))\), then there is an index \(i\) such that \(m_{A(i)}(\lambda )=m_A(\lambda )+1\). The authors prove that this result is generally true only for trees.