2025/08/14 by Nguyen, Tung T., Tân, Nguyen Duy
#05C25 #11L03 #11T24 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.10348
We define super-Cayley graphs over a finite abelian group G. Using the theory of supercharacters on G, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we show that a super-Cayley graph is determined by its spectrum once an indexing on the underlying group G is fixed. This generalizes a theorem by Sander-Sander, which investigates the case where G is a cyclic group. We then use our theory to define and study the concept of a U-unitary Cayley graph over a finite commutative ring R, where U is a subgroup of the unit group of R. Furthermore, when the underlying ring is a Frobenius ring, we show that there is a natural supercharacter theory associated with U. By applying the general theory of super-Cayley graphs developed in the first part, we explore various spectral properties of these U-unitary Cayley graphs, including their rationality and connections to various arithmetical sums.