# Spectral Symmetry in Conference Matrices

Willem H. Haemers, Leila Parsaei Majd

Research output: Working paperOther research output

### 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 Ithaca Cornell University Library Published - 13 Apr 2020

### Publication series

Name arXiv 2004.05829

• math.CO
• 05C50

• ## 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