Partial difference sets and amorphic Cayley schemes in non-abelian $2$-groups

TL;DR

This paper explores the construction of amorphic non-abelian Cayley schemes from abelian schemes within specific graph families, expanding understanding of automorphism groups in algebraic graph theory.

Contribution

It introduces methods to derive non-abelian regular automorphism groups from abelian ones in certain graph families, leading to new amorphic Cayley schemes.

Findings

Established conditions for non-abelian automorphism groups

Constructed new amorphic non-abelian Cayley schemes

Extended the theory of automorphism groups in graph families

Abstract

In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.