Spectral Symmetry in Conference Matrices

Willem H. Haemers, Leila Parsaei Majd

Research output: Working paperOther research output


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 languageEnglish
Place of PublicationIthaca
PublisherCornell University Library
Publication statusPublished - 13 Apr 2020

Publication series



  • math.CO
  • 05C50


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