A doodle of mine to help organize my thoughts. Here, each “edge” in the graph connects “vertices” that represent a triad. The triads do not represent every possible set of 3 notes, but they do represent every possible set of 3 notes where every note is “consonant” with at least one other note in the set, in the sense of being a perfect fourth or fifth, or a major or minor third (or major/minor sixth as you choose octaves). This is a very limited view of “consonant” but it’s a place to start. One can also think of this as being based on the 5-limit major scale. In this context, tracing paths through this graph shows possible harmonic progressions where each changes by exactly one note. If you manage to path through every possible edge exactly once, this is an Eulerian path. If you path through every possible vertex exactly once and end where you start, this is a cycle.
This idea can be altered in many ways to create other graphs: use 4-note chords instead of 3-note chords; change what constitutes “consonant”; change exactly 2 notes instead of 1; use different scales; etc.