Bordered complex Hadamard matrices and strongly regular graphs

TL;DR

This paper investigates bordered complex Hadamard matrices linked to strongly regular graphs, proving that only specific known examples exist within this framework, thereby classifying such matrices.

Contribution

The paper establishes a classification result showing no other bordered complex Hadamard matrices with cores in the Bose-Mesner algebra of strongly regular graphs exist beyond known examples.

Findings

Confirmed the uniqueness of certain bordered complex Hadamard matrices

Proved non-existence of additional matrices with the specified properties

Extended understanding of the relationship between Hadamard matrices and strongly regular graphs

Abstract

We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Sz\"{o}ll\H{o}si, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.