The Inverse Eigenvalue Problem for Linear Trees

2019/06/14 by Tanay Wakhare, Wakhare, Tanay, Charles R. Johnson +1 · 1 citation
Computer Science · Mathematics · #05C50 #15B57 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Mathematical Approximation and Integration #Matrix Theory and Algorithms #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1906.06257

openalex publication_date 2019/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014. This is the most general class of trees for which the inverse eigenvalue problem has been solved. We explore many consequences, including the Degree Conjecture for possible spectra, upper bounds for the minimum number of eigenvalues of multiplicity 1, and the equality of the diameter of a linear tree and its minimum number of distinct eigenvalues, etc.

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