Construction of matrices with a given graph and prescribed interlaced spectral data

A result of Duarte [Linear Algebra Appl. 113 (1989) 173–182] asserts that for real λ1,…,λnλ1,…,λn, μ1,…,μn-1μ1,…,μn-1 withλ1<μ1<λ2<μ2<⋯<μn-1<λn,λ1<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,…,λnλ1,λ2,…,λn and the principal submatrix obtained from A   by deleting its last row and column has eigenvalues μ1,…,μn-1μ1,…,μn-1. This result is extended to connected graphs through the use of the implicit function theorem.