Fourier analysis on distance-regular Cayley graphs over abelian groups

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.