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 $U_{R,k}=\{x^k:x\in R^{\ast }\}$, 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 $\mathfrak{m}$, we give the blow-up decompositions for the graphs: namely, we have $G_R(k)=(G_{R/\mathfrak{m}}(k){)}^{(|\mathfrak{m}|)}$ and ${\mathcal{G}}_R(k)=({\mathcal{G}}_{R/\mathfrak{m}}(k){)}^{(|\mathfrak{m}|)}$ 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 $G_{R_i}(k)=\Gamma (k_i,q_i)$ and similarly for ${\mathcal{G}}_{R_i}(k)$, for $i=1,\dots ,s$. Also, the reduced graphs correspond to the graphs of the reduced rings, i.e.\@ $(G_R(k){)}_{red}\simeq G_{R_{red}}(k)$ and $({\mathcal{G}}_R(k){)}_{red}\simeq {\mathcal{G}}_{R_{red}}(k)$. By using these decompositions in terms of GP-graphs, we study some basic structural properties of the graphs such as directedness, bipartiteness and connectedness.

arXiv comment39 pages, 2 figures