Fourier analysis on distance-regular Cayley graphs over abelian groups

TL;DR

The paper addresses the problem of characterizing distance-regular Cayley graphs over abelian groups, focusing on how Fourier analysis on finite abelian groups interfaces with finite geometry. By linking the distance module to Schur rings and exploiting character sums, the authors derive algebraic conditions that a Cayley graph must satisfy to be distance-regular, and they leverage these to control possible structures in the abelian setting. They then integrate finite geometry—relative difference sets and transversal designs—to realize and classify specific graph families as line graphs and other standard constructions, culminating in a complete classification for graphs on $\mathbb{Z}_n \oplus \mathbb{Z}_p$ with odd prime $p$ dividing $n$. The main result identifies four explicit families, including a line-graph realization of transversal designs, and clarifies when these graphs are primitive versus imprimitive, thereby connecting spectral, combinatorial, and geometric perspectives in a unified framework. This classification advances understanding of how distance-regularity manifests in Cayley graphs over abelian groups and highlights the role of finite geometry in producing concrete examples with classical parameters.

Abstract

The problem of constructing or characterizing strongly regular Cayley graphs (or equivalently, regular partial difference sets) has garnered significant attention over the past half-century. In 2003, Miklavič and Potočnik [European J. Combin. 24 (2003) 777--784] expanded upon this field by achieving a complete characterization of distance-regular Cayley graphs over cyclic groups through the method of Schur rings. Building on this work, Miklavič and Potočnik [J. Combin. Theory Ser. B 97 (2007) 14--33] formally proposed the problem of characterizing distance-regular Cayley graphs for arbitrary classes of groups. Within this framework, abelian groups hold particular significance, as numerous distance-regular graphs with classical parameters are precisely Cayley graphs over abelian groups. In this paper, we employ Fourier analysis on abelian groups to establish connections between distance-regular Cayley graphs over abelian groups and combinatorial objects in finite geometry. By combining these insights with classical results from finite geometry, we classify all distance-regular Cayley graphs over the group $\mathbb{Z}_n \oplus \mathbb{Z}_p$, where $p$ is an odd prime.

Theorems & Definitions (44)

• Theorem 1.1: MP03
• Theorem 1.2
• Lemma 2.1: MP09
• Lemma 2.2: KT
• Lemma 2.3: Bannai
• Lemma 2.4: BCN89
• Lemma 2.5: BCN89
• Lemma 2.6: BCN89
• Lemma 2.7
• proof