Generalized spectral characterization of signed trees

TL;DR

This paper proves that under certain algebraic conditions on a tree's characteristic polynomial, all signed trees with that underlying graph are uniquely identified by their generalized spectrum.

Contribution

It establishes a new spectral characterization criterion for signed trees based on the discriminant of their characteristic polynomial.

Findings

If $2^{-rac n2}\sqrt{ riangle(T)}$ is odd and square-free, then signed trees are determined by their generalized spectrum.

The criterion links algebraic properties of the characteristic polynomial to spectral uniqueness.

The result applies to trees with irreducible characteristic polynomials over $Q$.

Abstract

Let $T$ be a tree with an irreducible characteristic polynomial $\phi(x)$ over $\mathbb{Q}$. Let $\Delta(T)$ be the discriminant of $\phi(x)$. It is proved that if $2^{-\frac n2}\sqrt{\Delta(T)}$ (which is always an integer) is odd and square free, then every signed tree with underlying graph $T$ is determined by its generalized spectrum.