Fusions of the generalized Hamming scheme on a strongly-regular graph

TL;DR

This paper explores how fusions of association schemes influence the structure of generalized Hamming schemes, especially on strongly-regular graphs, revealing new fusion possibilities and parameter conditions.

Contribution

It demonstrates that fusions of association schemes induce nontrivial fusions in generalized Hamming schemes and characterizes strongly-regular graphs with extra fusions in their schemes.

Findings

Generalized Hamming schemes inherit fusions from underlying association schemes.

Identifies parameters of strongly-regular graphs with additional scheme fusions.

Provides conditions for extra fusions beyond trivial ones.

Abstract

In this paper we show that for any fusion $\mathcal{B}$ of an association scheme $\mathcal{A}$, the generalized Hamming scheme $H(n,\mathcal{B})$ is a nontrivial fusion of $H(n,\mathcal{A})$. We analyze the case where $\mathcal{A}$ is the association scheme on a strongly-regular graph, and determine the parameters of all strongly-regular graphs for which the generalized Hamming scheme, $H(2,\mathcal{A})$, has extra fusions, in addition to the one arising from the trivial fusion of $\mathcal{A}$.