I don’t think it will be possible to have a very huge number of neighbouring biomes. So if the graph is circles that are on a 2d plane and can move freely it should be possible to clearly display a whole planet that has a reasonable number of smallish biomes. Perhaps looking like this:
For that we can represent that as weights on the vertices between the biomes, and a higher weight would indicate that it’s easier to move to that biome, which would be calculated from the amount of shared border, currents etc. Basically I want the biomes to be a mathematical (unfortunately the wikipedia article has no graphs with weights) graph that is displayed graphically to the user.