Hadamard diagonalizable graphs of order at most 36

TL;DR

This paper characterizes Hadamard diagonalizable graphs of small order, proving specific forms for certain sizes and providing methods to identify all such graphs up to order 36, despite the unknown number of Hadamard matrices at that order.

Contribution

It establishes classification results for Hadamard diagonalizable graphs of order 8k+4 and develops an efficient method to find all such graphs for any Hadamard matrix.

Findings

Only specific graphs are Hadamard diagonalizable when order is 8k+4.

All Hadamard diagonalizable graphs up to order 36 are identified.

A new computational method for diagonalization by Hadamard matrices is introduced.

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.