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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 $$G= (V,E)$$.

##### MSC:
 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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