Fourier analysis on distance-regular Cayley graphs over abelian groups

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, Miklavi\v{c} and Poto\v{c}nik [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, Miklavi\v{c} and Poto\v{c}nik [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.