Hermitian matrices of roots of unity and their characteristic polynomials

2021/06/10 by Gary R. W. Greaves, Greaves, Gary R. W., Chin Jian Woo +1
Mathematics · Chemistry · #Graph theory and applications #Molecular spectroscopy and chirality #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2106.05477

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-ζ), where ζ 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.

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