Spectral properties of Cayley graphs over finite commutative rings

TL;DR

This paper extends spectral results for unitary Cayley graphs to Cayley graphs ${\rm Cay}(R,xR^{*})$ over finite commutative rings by deriving the full spectrum and energy, as well as the energy of the complement, in terms of local ring invariants. The authors establish that ${\rm Cay}(R,xR^{*})$ decomposes via a local-ring tensor-product framework, yielding explicit eigenvalues $(-1)^{|C|}\frac{|xR^{*}|}{\prod_{i\in C}(\frac{|I_{x_i}|}{|M_{x_i}|}-1)}$ with computable multiplicities, and a compact energy formula $\mathcal{E}=2^{|P|}\frac{|R||xR^{*}|}{|I_x|}$. They further derive a closed expression for the energy of the complement graph and provide a thorough set of necessary and sufficient conditions for ${\rm Cay}(R,xR^{*})$ to be Ramanujan, expressed through local invariants $\frac{I_{x_i}}{M_{x_i}}$ and related ring-theoretic data. The Ramanujan characterization includes numerous explicit cases and a reduction principle to a smaller ring formed by the active coordinates, demonstrating when nontrivial eigenvalues meet the optimal spectral gap bound $2\sqrt{|xR^{*}|-1}$.

Abstract

Let $R$ be a finite commutative ring with unity and $x$ be a non-zero element of $R$. In this paper, we calculate the spectrum and energy of the Cayley graph ${\rm Cay}(R,xR^{*})$, and also compute the energy of their compliment graph. Further, we give necessary and sufficient condition for Cayley graph ${\rm Cay}(R,xR^{*})$ to be Ramanujan.

Theorems & Definitions (31)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3