A region-based low-carbon operation analysis method for integrated electricity‑hydrogen-gas systems (IEHGSs) is proposed in this paper. The committed carbon emission operation region (CCE_OR) and the economic CCE_OR (ECCE_OR) are proposed to evaluate the low-carbon feasible space (LCFS) from the low-carbon, secure, and economic perspectives, which provides powerful tools for the low-carbon operation analyses of IEHGSs. Firstly, a coordinated mechanism of carbon flow and hydrogen flow is developed to excavate the low-carbon operation potential of IEHGSs. Then, based on the distributionally robust chance-constrained approach, the concepts and models of CCE_OR and ECCE_OR are defined and formulated considering uncertainties and hydrogen merits. Furthermore, a region model reformulation method is proposed to reformulate the uncertain and non-convex CCE_OR and ECCE_OR models into the deterministic and mixed-integer linear region models. The geometric properties of CCE_OR and ECCE_OR are both theoretically proven, which are the union set of finite convex and bounded polytopes. The united region solution methodology considering the geometric properties of CCE_OR and ECCE_OR is presented to effectively characterize CCE_OR and ECCE_OR. Finally, the numerical results on two test systems verify the effectiveness and applicability of the proposed methods. Simulation results indicate that the proposed methods can improve the LCFS, characterize visual coupling relationships between LCFS and observation variables, and provide comprehensive low-carbon operation information of operation points.