In this paper, aiming to systematically derive single-inductor multi-input multi-output (SI-MIMO) converters having favorable merits of high-density and low-cost, a basic model is firstly extracted from the working principle. Theoretically, by enumerating all possible connections of the basic model and one-by-one manually judging their effectiveness according to working criterions, multiple viable SI-MIMO topologies can be obtained, but with heavy workload. To simplify the topology derivation process, this paper also proposes a computer-aided method to replace the manual effort. By modelling the basic model and working criterions with graph theory, MATLAB codes can be developed to automatically screen out the effective topologies. With the proposed method, 8 three-port, 28 four-port and 115 five-port converters with single inductor are conveniently obtained, including both the existing and new topologies. Moreover, by configuring ports and switches in each derived topology, multiple specific converters can be further expanded. These converters can provide more options for practical engineering applications, and a new topology is also introduced and experimentally verified in the paper.