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A result of Duarte [Linear Algebra Appl. 113 (1989) 173–182] asserts that for real λ1,…,λn, μ1,…,μn-1 withλ1<μ1<λ2<μ2<⋯<μn-1<λn,and each tree T on n vertices there exists an n×n, real symmetric matrix A whose graph is T such that A has eigenvalues λ1,λ2,…,λn and the principal submatrix obtained from A by deleting its last row and column has eigenvalues μ1,…,μn-1. This result is extended...
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