I need a big regular polytope!

Question

I have an ingenious project in mind that requires me to begin with a “big” graph of a regular polytope (# of vertices >, say, 50? 100?). This polytope can be of any dimension (I won’t be building it or drawing it). It only needs to be “regular” in the sense that each vertex has the same number of edges. That is, what I want is the chart that lists the connectivity between the vertices (I don’t even need their coordinates in whatever dimensional space this thing lives in). That is all I need. Ideally I would like a big polytope with five edges meeting at every vertex and as many edges as I can get. However, I can work with other vertex structures if I must. 1) Does such a thing even exist? 2) Is there a database of such polytopes with the vertex chart structure that I need?

Answer 1

The House of Graphs has a searchable database of "interesting" graphs; you could try searching for large $5$-regular planar graphs (planarity would take care of your requirement that the graph comes from a polytope, if I'm correctly understanding that your "regularity" requirement doesn't need any algebraic symmetry). You could also try using plantri to generate planar graphs, although some more assembly might be required there.