# The Inverse Eigenvalue Problem for Linear Trees

## TL;DR

This paper proves the sufficiency of the Linear Superposition Principle for the inverse eigenvalue problem in linear trees, establishing the most general class of trees with a solved spectral characterization.

## Contribution

It demonstrates the sufficiency of the Linear Superposition Principle for linear trees and explores several spectral properties and conjectures related to this class.

## Key findings

- Sufficiency of the Linear Superposition Principle for linear trees.
- Validation of the Degree Conjecture for these spectra.
- Establishment of bounds on eigenvalue multiplicities and relations to tree diameter.

## 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.