2024/07/10 by Zhan, Xiongfeng, Huang, Xueyi, Lu, Lu
#05C25 #05C50 #05E30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.08763
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 ℤn ⊕ ℤp, where p is an odd prime.