A generalization of Ramanujan's sum over finite rings

2025/05/15 by Priya, Singh, Sanjay Kumar
#05C25 #05C50 #20C15 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.09996

Abstract

Let R be a finite ring with unity. In general, the eigenvalues of the unitary Cayley graph Cay(R, R^×) are not known when R is a non-commutative. In this paper, we present an explicit formula for the eigenvalues of Cay(R, R^×) for any finite ring R. However, our focus is on a more general case of the unitary Cayley graph. It is well known that the classical Ramanujan's sum represents the eigenvalues of Cay(ℤ_n, ℤ_n^×). Consequently, the eigenvalues of Cay(R, R^×) can be view as a generalization of classical Ramanujan's sum in the context of finite rings. Interestingly, the formula we derive for the eigenvalues of Cay(R, R^×) extends the known formula of classical Ramanujan's sum to the context of finite rings.

Citations

• Spectral properties of Cayley graphs over finite commutative rings

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