Hadamard diagonalizable graphs of order at most 36

Date
2020-07-17
Authors
Breen, Jane
Lidicky, Bernard
Butler, Steve
Fuentes, Melissa
Lidicky, Bernard
Phillips, Michael
Riasanovsky, Alexander
Song, Sung-Yell
Villagrán, Ralihe
Wiseman, Cedar
Zhang, Xiaohong
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Mathematics
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Mathematics
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.

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

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