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arXiv · 2004.05829

Spectral symmetry in conference matrices

Willem H. Haemers·Leila Parsaei Majd
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Abstract

A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $\Sigma$ (that is, $\Sigma=-\Sigma$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal submatrices of $C$ also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.

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Willem H. Haemers, Leila Parsaei Majd. 2020-04-13. Spectral symmetry in conference matrices. https://arxiv.org/abs/2004.05829

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