The inverse eigenvalue problem for linear trees

We prove the sufficiency of Linear Superposition Principle for linear trees, which characterizes spectra achievable by a real symmetric matrix whose underlying graph is tree. The necessity was previously proven in [10] . This most general class trees inverse eigenvalue problem has been solved. explore many consequences, including Degree Conjecture possible spectra, upper bounds minimum number eigenvalues multiplicity 1, and equality diameter tree its distinct eigenvalues.