Quantitative structure property research works, which are the essential part in chemical information and modelling, give basic underlying topological properties for chemical substances. This information enables conducting more feasible studies between theory and practice. Connectivity concept in chemical graph theory gives information about underlying topology of chemical structures, fault tolerance of molecules, and vulnerability of chemical networks. In this study we first defined two novel types of conditional connectivity measures based on regularity notion: k-regular edge connectivity and almost k-regular edge connectivity in chemical graph theory literature. We computed these new graph invariants for cycles, complete graphs, and Cartesian product of cycles. Our results will be applied to calculate k-regular edge connectivity of some nanotubes which are stated as Cartesian product of cycles. These calculations give information about fault tolerance capacity and vulnerability of these chemical structures.