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

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

2010-11-11

arxiv.org/abs/1011.2737v3

Mathematics

Number Theory

Scientific paper

10.1016/j.jalgebra.2011.02.009

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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