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Selected Works of Bernard Lidický
Article
Hadamard diagonalizable graphs of order at most 36
arXiv
  • Jane Breen, Ontario Tech University
  • Steve Butler, Iowa State University
  • Melissa Fuentes, University of Delaware
  • Bernard Lidicky, Iowa State University
  • Michael Phillips, University of Colorado, Denver
  • Alexander W. N. Riasanovsky, Iowa State University
  • Sung-Yell Song, Iowa State University
  • Ralihe R. Villagrán, Centro de Investigación y de Estudios Avanzados del IPN
  • Cedar Wiseman, University of Wyoming
  • Xiaohong Zhang, University of Waterloo
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Document Type
Article
Disciplines
Publication Version
Submitted Manuscript
Publication Date
7-17-2020
Abstract

If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries ±1, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable.
In this article, we prove that if n=8k+4 the only possible Hadamard diagonalizable graphs are Kn, Kn/2,n/2, 2Kn/2, and nK1, and we develop an efficient computation for determining all graphs diagonalized by a given Hadamard matrix of any order. Using these two tools, we determine and present all Hadamard diagonalizable graphs up to order 36. Note that it is not even known how many Hadamard matrices there are of order 36.

Comments

This preprint is made available through arXiv: https://arxiv.org/abs/2007.09235.

Copyright Owner
The Authors
Copyright Date
2020
Language
en
File Format
application/pdf
Citation Information
Jane Breen, Steve Butler, Melissa Fuentes, Bernard Lidicky, et al.. "Hadamard diagonalizable graphs of order at most 36" arXiv (2020)
Available at: http://works.bepress.com/bernard-lidicky/65/