How would one find the number of spanning trees in a unlabled graph

Question

How would I go about finding the number of spanning trees? Is there a techunique to do this?

Answer 1

For an arbitrary graph, you would use something like Kirchhoff's theorem. But with a graph like yours with lots of symmetry one may hope to find a clever shortcut specific to that graph.

For example, consider the 9 "outer" edges of the graph, and consider which of them are not in the spanning tree. It is clear that at most one of them can be missing in each of the squares, or some vertices would be cut off. So we can do an analysis by cases:

• If none of the 9 outer edges are missing, then they form a cycle. Impossible.

• If one of the 9 outer edges are missing, then the remaining 8 form a spanning tree together.

• If two of the 9 outer edges are missing, they must be in different squares, and exactly one of the corresponding "inner" eges must be added to the spanning tree.

• If three of the 9 outer edges are missing, then we must choose exactly two of the inner edges to complete the tree.

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Alternatively: Exactly one of the three corners of the triangle must have the property that the path between the two other corners (along the spanning tree) passes through it.

Each of the squares adjacent to that corner can have a spanning tree selected in $4$ ways, and then there are $3$ ways to connect the two outer vertices in the last square.