A Note on the Component (Extra) Edge Connectivity of the Cartesian Powers of Regular Multiprocessor Systems

Reliability assessment of multiprocessor systems presents the theoretical foundation for the layout and optimization of multiprocessor systems. The $h$-extra edge-connectivity $\lambda _{h}$ and the $k$-component edge-connectivity $c\lambda _{k}$, as extensions of the classical edge connectivity, are two precise metrics for the measurement of the reliability of multiprocessor systems. For multiprocessor systems, determining $c\lambda _{k}$ and $\lambda _{k}$ of a large $k$ is still difficult. Let $\delta _{G}(0)=0$ and $\delta _{G}(i)=\frac{1}{2}(\text{ex}_{i+1}(G)-\text{ex}_{i}(G))$ for $i\in \lbrace 1, \ldots, |G|-1\rbrace$, where $\text{ex}_{i}(G)={\mathrm{max}}\lbrace 2|E(G[S])|: S\subseteq V(G), |S|=i \rbrace$. In this article, we obtain $c\lambda _{k}$ and $\lambda _{h}$ of the Cartesian powers of the $d$-regular graphs for which the lexicographic order yields an optimal order and $\delta _{G}(i)\leq \frac{d}{2}$ for $i=0, 1, \ldots, \lfloor \frac{|G|-1}{2}\rfloor$. Our result improves some previous results about $c\lambda _{k}$ of Hamming graphs by Yang et al. (2023), and $\lambda _{h}$ of the Cartesian powers of the complete graph $K_{4}$ by Tian et al. (2022).