Isomorphism classes of association schemes induced by Hadamard matrices

TL;DR

This paper investigates the classification of association schemes derived from Hadamard matrices, focusing on their isomorphism classes and conditions for fission schemes, contributing to algebraic combinatorics and graph theory.

Contribution

It introduces methods to estimate the number of isomorphism classes of fission schemes with given intersection numbers derived from Hadamard matrices.

Findings

Estimation of the number of isomorphism classes for certain fission schemes

Conditions under which fission schemes are induced by Hadamard matrices

Analysis of the structure of association schemes from Hadamard graphs

Abstract

Every Hadamard matrix $H$ of order $n > 1$ induces a graph with $4n$ vertices, called the Hadamard graph $\Gamma(H)$ of $H$. Since $\Gamma(H)$ is a distance-regular graph with diameter $4$, it induces a $4$-class association scheme $(\Omega, S)$ of order $4n$. In this article we deal with fission schemes of $(\Omega, S)$ under certain conditions, and for such a fission scheme we estimate the number of isomorphism classes with the same intersection numbers as the fission scheme.