Switching graphs and Hadamard matrices

TL;DR

This paper establishes an equivalence between two switching methods used to construct inequivalent Hadamard matrices and cospectral graphs, linking two previously separate approaches in combinatorial structure transformations.

Contribution

It demonstrates that two different switching techniques for constructing inequivalent Hadamard matrices and cospectral graphs are fundamentally equivalent.

Findings

Shows equivalence between two switching methods for Hadamard matrices and graphs.

Bridges the gap between Orrick's and Godsil-McKay's switching techniques.

Provides a unified perspective on combinatorial structure transformations.

Abstract

Local operations of combinatorial structures (graphs, Hadamard matrices, codes, designs) that maintain the basic parameters unaltered, have been widely used in the literature under the name of switching. We show an equivalence between two switching methods to construct inequivalent Hadamard matrices, which were proposed by Orrick [SIAM Journal on Discrete Mathematics, 2008], and the switching method for constructing cospectral graphs which was introduced by Godsil and McKay [Aequationes Mathematicae, 1982].