"Find a graph with 8 vertices, which have exactly 27 spanning trees." How do I find such a graph, or prove one does not exist?
2026-03-30 07:11:30.1774854690
Graph with fixed amount of spanning trees
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27 = 3*3*3, so maybe we have to work with triangles. A triangle has three spanning trees. Let's take three triangles and try to connect them in such a way that we get 27 spanning trees and 8 edges.
Call them triangle A, B, C.
Connect triangle A and B together so that they share a vertex. Now we have 8 vertices. Connect a vertex from triangle C to the shared vertex of triangle A and B through an edge. Now we have 3*3*3 spanning trees.
I realize my description might be confusing so this may be more clear: