Geodesic
Hadamard diagonalizable graphs of order at most 36
Jane Breen1 ; Steve Butler2 ; Melissa Fuentes3 ; Bernard Lidický2 ; Michael Phillips4 ; Alexander Riasanovksy2 ; Sung-Yell Song2 ; Ralihe Villagrán5 ; Cedar Wiseman6 ; Xiaohong Zhang7
1Ontario Tech University
2Iowa State University
3University of Delaware
4University of Colorado Denver
5Centro de Investigación u de Estudios Avanzados del IPN
6University of Wyoming
7University of Waterloo
The electronic journal of combinatorics, Tome 29 (2022) no. 2
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If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $\pm1$, 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 $K_n$, $K_{n/2,n/2}$, $2K_{n/2}$, and $nK_1$, and we develop a computational method 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.
DOI : 10.37236/9725
Classification : 05C50, 15B34, 05B20, 05C76, 05C85
Mots-clés : Laplacian matrix of a graph, Hadamard matrix, Hadamard diagonalizable graph

Jane Breen  1   ; Steve Butler  2   ; Melissa Fuentes  3   ; Bernard Lidický  2   ; Michael Phillips  4   ; Alexander Riasanovksy  2   ; Sung-Yell Song  2   ; Ralihe Villagrán  5   ; Cedar Wiseman  6   ; Xiaohong Zhang  7

1 Ontario Tech University
2 Iowa State University
3 University of Delaware
4 University of Colorado Denver
5 Centro de Investigación u de Estudios Avanzados del IPN
6 University of Wyoming
7 University of Waterloo 
@article{10_37236_9725,
     author = {Jane Breen and Steve Butler and Melissa Fuentes and Bernard Lidick\'y and Michael Phillips and Alexander Riasanovksy and Sung-Yell Song and Ralihe Villagr\'an and Cedar Wiseman and Xiaohong Zhang},
     title = {Hadamard diagonalizable graphs of order at most 36},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {2},
     doi = {10.37236/9725},
     zbl = {1487.05153},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9725/}
}
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AU  - Alexander Riasanovksy
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Jane Breen; Steve Butler; Melissa Fuentes; Bernard Lidický; Michael Phillips; Alexander Riasanovksy; Sung-Yell Song; Ralihe Villagrán; Cedar Wiseman; Xiaohong Zhang. Hadamard diagonalizable graphs of order at most 36. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/9725

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