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. A classic result in this area is the complete classification of strongly regular Cayley graphs over cyclic groups, which was established by Bridges and Mena (1979), independently by Ma (1984), and partially by Marušič (1989). Miklavič and Potočnik (2003) extended this work by providing a complete characterization of distance-regular Cayley graphs over cyclic groups through the method of Schur rings. Building on this, Miklavič and Potočnik (2007) formally posed the problem of characterizing distance-regular Cayley graphs for arbitrary classes of groups. Within this framework, abelian groups are of particular significance, as many distance-regular graphs with classical parameters are 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 $n$ is a positive integer and $p$ is an odd prime.