Hermitian matrices of roots of unity and their characteristic polynomials

TL;DR

This paper explores spectral properties of Hermitian matrices with roots of unity, establishing conjecturally sharp bounds on their characteristic polynomial residues and generalizing classical trace relations for graph adjacency matrices.

Contribution

It introduces conjecturally sharp bounds on characteristic polynomial residues and generalizes a classical trace relation, advancing understanding of spectral properties of root-of-unity Hermitian matrices.

Findings

Conjecturally sharp upper bounds on polynomial residue classes.

A generalized relation for traces of powers of graph-adjacency matrices.

New spectral conditions for Hermitian matrices with roots of unity.

Abstract

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of $(1-\zeta)$, where $\zeta$ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results.