The conventional topology derivation of common-grounded switched-capacitor multilevel inverter (CGSCMLI) topologies relies on manual effort, which is complex and inefficient. This paper presents a programmable method for CGSCMLIs based on graph theory. By mathematically modeling CGSCMLIs with graph theory and translating their working criterion into graph constraints, the topology derivation can be automated and streamlined through a computer program. In this paper, the principle of the proposed graph-theory-based programmable method is introduced first. Subsequently, it is applied to derive all effective 5-level CGSCMLIs (CG5LSCMLIs), discovering a variety of new topologies. The topology with minimum semiconductor count and minimum voltage stress is selected automatically by the program for theoretical analysis and simulation validation.