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arXiv · 1011.2737

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

Graeme Taylor
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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 $\pm2$.

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Graeme Taylor. 2011-02-08. Cyclotomic Matrices and Graphs over the ring of integers of some imaginary quadratic fields. https://doi.org/10.1016/j.jalgebra.2011.02.009

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