Hadamard diagonalizable graphs of order at most 36
Abstract
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 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.09235
 Bibcode:
 2020arXiv200709235B
 Keywords:

 Mathematics  Combinatorics;
 05C50;
 15B34;
 05B20;
 05C76;
 05C85