Cyclotomic Matrices and Graphs over the ring of integers of some imaginary quadratic fields

TL;DR

This paper classifies Hermitian matrices over certain imaginary quadratic integer rings with eigenvalues in [-2,2], extending graph-theoretic methods to these algebraic settings and identifying maximal configurations.

Contribution

It generalizes charged signed graphs to $ ext{L}$-graphs over specific quadratic fields and classifies all with eigenvalues in [-2,2], revealing maximal structures.

Findings

Complete classification of Hermitian matrices with eigenvalues in [-2,2] over specified rings.

Introduction of $ ext{L}$-graphs as a generalization of charged signed graphs.

Identification of maximal matrices/graphs with eigenvalues $ ext{±}2$.

Abstract

We determine all Hermitian $\mathcal{O}_{\Q(\sqrt{d})}$-matrices for which every eigenvalue is in the interval [-2,2], for each d in {-2,-7,-11,-15\}. To do so, we generalise charged signed graphs to $\mathcal{L}$-graphs for appropriate finite sets $\mathcal{L}$, and classify all $\mathcal{L}$-graphs satisfying the same eigenvalue constraints. We find that, as in the integer case, any such matrix / graph is contained in a maximal example with all eigenvalues $\pm2$.