### 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.

Original language | English |
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Place of Publication | Ithaca |

Publisher | Cornell University Library |

Publication status | Published - 13 Apr 2020 |

### Publication series

Name | arXiv |
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Volume | 2004.05829 |

### Keywords

- math.CO
- 05C50

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## Cite this

Haemers, W. H., & Parsaei Majd, L. (2020).

*Spectral Symmetry in Conference Matrices*. (arXiv; Vol. 2004.05829). Cornell University Library. https://arxiv.org/abs/2004.05829