Spectral symmetry in conference matrices

Willem H. Haemers; Leila Parsaei Majd

doi:10.1007/s10623-021-00858-8

Spectral symmetry in conference matrices

Willem H. Haemers^*, Leila Parsaei Majd

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

A conference matrix of order n is an n× n matrix C with diagonal entries 0 and off-diagonal entries ± 1 satisfying CC^⊤= (n- 1) I. If C is symmetric, then C has a symmetric spectrum Σ (that is, Σ = - Σ) and eigenvalues ±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
Journal	Designs, Codes, and Cryptography
DOIs	https://doi.org/10.1007/s10623-021-00858-8
Publication status	Accepted/In press - 2021

Keywords

Conference matrix
Paley graph
Seidel matrix
Signed graph
Symmetric spectrum

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title = "Spectral symmetry in conference matrices",

abstract = "A conference matrix of order n is an n× n matrix C with diagonal entries 0 and off-diagonal entries ± 1 satisfying CC⊤= (n- 1) I. If C is symmetric, then C has a symmetric spectrum Σ (that is, Σ = - Σ) and eigenvalues ±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.",

keywords = "Conference matrix, Paley graph, Seidel matrix, Signed graph, Symmetric spectrum",

author = "Haemers, {Willem H.} and {Parsaei Majd}, Leila",

note = "Funding Information: The research of the second author was funded by Iranian National Science Foundation (INSF) under the contract No. 98021291. Publisher Copyright: {\textcopyright} 2021, The Author(s). Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

year = "2021",

doi = "10.1007/s10623-021-00858-8",

language = "English",

journal = "Designs, Codes and Cryptography",

issn = "0925-1022",

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AU - Parsaei Majd, Leila

N1 - Funding Information: The research of the second author was funded by Iranian National Science Foundation (INSF) under the contract No. 98021291. Publisher Copyright: © 2021, The Author(s). Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - A conference matrix of order n is an n× n matrix C with diagonal entries 0 and off-diagonal entries ± 1 satisfying CC⊤= (n- 1) I. If C is symmetric, then C has a symmetric spectrum Σ (that is, Σ = - Σ) and eigenvalues ±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.

AB - A conference matrix of order n is an n× n matrix C with diagonal entries 0 and off-diagonal entries ± 1 satisfying CC⊤= (n- 1) I. If C is symmetric, then C has a symmetric spectrum Σ (that is, Σ = - Σ) and eigenvalues ±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.

KW - Conference matrix

KW - Paley graph

KW - Seidel matrix

KW - Signed graph

KW - Symmetric spectrum

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DO - 10.1007/s10623-021-00858-8

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