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

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 $\pm 2$.

References in corpus (3)

• A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices
• Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
• Integer symmetric matrices of small spectral radius and small Mahler measure