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On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices. (English) Zbl 1055.15010
Negative answers to the following two questions are given: 1. Is \(\{\lambda_1<\cdots< \lambda_k\}\) with multiplicities \(m_1,\dots, m_k\) the spectrum of some \(A\) in \(S(G)\), \(G\) a tree? 2. Is the minimum number of distinct eigenvalues over all \(A\) in \(S(G)\), \(G\) a tree, equal to the dimension of \(G\) plus 1? Here \(S(G)\) is the set of symmetric \(n\times n\) matrices \(A\) such that the graph \(G(A)\) of \(A\) equals a given graph \(G= (V,E)\).

15A18 Eigenvalues, singular values, and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A29 Inverse problems in linear algebra
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