On $k$-th unitary Cayley graphs over finite commutative rings: structure and decompositions

Abstract

Given $R$ a finite commutative ring with identity and $k \in \mathbb{N}$, we consider the $k$-th unitary Cayley graph $G_R(k)=Cay(R,U_{R,k})$ with $U_{R,k} = \{ x^k: x \in R^*\}$, and its symmetrized version $\mathcal{G}_R(k) = Cay(R,T_{R,k})$, with $T_{R,k}=U_{R,k} \cup (-U_{R,k})$. If $R$ is a local ring with maximal ideal $\frak m$, we give the blow-up decompositions for the graphs: namely, we have $G_R(k)= (G_{R/\frak m}(k))^{(|\frak m|)}$ and $\mathcal{G}_R(k)= (\mathcal{G}_{R/\frak m}(k))^{(|\frak m|)}$ for any $k$ such that $(k,|R|)=1$. If the ring $R$ has Artin decomposition $R=R_1 \times \cdots \times R_s$ in local rings $R_i$, we give the Kronecker product decompositions $G_R(k) = G_{R_1}(k) \otimes \cdots \otimes G_{R_s}(k)$ and $\mathcal{G}_R(k) = \mathcal{G}_{R_1}(k) \otimes \cdots \otimes \mathcal{G}_{R_s}(k)$. In further $(k,|R|)=1$, these decompositions can be given in terms of generalized Paley (GP) graphs over finite fields, that is $G_{R_i}(k) = Γ(k_i,q_i)$ and similarly for $\mathcal{G}_{R_i}(k)$, for $i=1,\ldots,s$. Also, the reduced graphs correspond to the graphs of the reduced rings, i.e.\@ $\big(G_{R}(k)\big)_{red}\simeq G_{R_{red}}(k)$ and $\big(\mathcal{G}_{R}(k)\big)_{red} \simeq \mathcal{G}_{R_{red}}(k)$. By using these decompositions in terms of GP-graphs, we study some basic structural properties of the graphs such as directedness, bipartiteness and connectedness.