For a graph \(G\) , a \(k\) -regular edge cut is an edge subset \(S \subseteq E(G)\) such that \(G-S\) is disconnected and each connected component of \(G-S\) is a \(k\) -regular graph. The \(k\) -regular edge-connectivity is defined as the minimum cardinality among all \(k\) -regular edge cuts. The \(h\) -extra \(r\) -component edge-connectivity \(c\lambda _r^h(G)\) of \(G\) is the minimum cardinality among all edge subsets \(F \subseteq E(G)\) , if any, such that \(G-F\) has exactly \(r\) components and each component has at least \(h\) vertices. In this paper, we determine that the \(h\) -extra 5-component edge-connectivity of \({FQ}_n\) is \(c\lambda _5^h(FQ_{n})={4nh-4ex_h(FQ_n)}\) when \(n\ge 6\) and \(1\le h\le 2^{\lfloor \frac{n}{2}\rfloor -2}\) . In addition, we derive the \(k\) -regular edge-connectivities of the folded hypercubes \({FQ}_n\) and the hierarchical folded cubes \({HFQ}_n\) as follows: \(\lambda ^{kr}({FQ}_n) = 2^{n-1}(n+1-k)\) for \(n \ge 2\) and \(1 \le k \le n-1\) ; \(\lambda ^{kr}({HFQ}_n) = 2^{2n-1}(n+2-k)\) for \(n \ge 2\) and \(1 \le k \le n + 1\) .
