IGraph/M now includes a function to count spanning trees.

In[1]:= Needs["IGraphM`"]

During evaluation of In[1]:= IGraph/M 0.3.99.1 (May 1, 2018)

During evaluation of In[1]:= Evaluate IGDocumentation[] to get started.

In[2]:= IGSpanningTreeCount[HypercubeGraph[3]]
Out[2]= 384

To explore the topic, you might find it useful to generate some more random spanning trees and visualize them.

g = HypercubeGraph[3, VertexLabels -> Automatic]

IGRandomSpanningTree[g, 20] // 
  DeleteDuplicatesBy[AdjacencyMatrix] // 
  Map[HighlightGraph[g, #, GraphHighlightStyle -> "Thick"] &]

While this graph has 384 distinct labelled spanning trees, only 6 of them are non-isomorphic.

IGRandomSpanningTree[HypercubeGraph[3], 100] // 
 DeleteDuplicatesBy[CanonicalGraph]

The output of 384 refers to the number of spanning trees in the 3-hypercube graph. A spanning tree is basically a connected acyclic subset of edges in the graph such that each vertex in the graph is incident on at least one edge in the subset. (Kirchoff's Theorem gives a very nice and efficient algorithm for computing the number of spanning trees, and Combinatorica.m uses that approach.)

You can use Combinatorica's MinimumSpanningTree to get at least one MST. To get other ones, you could delete an edge you would like to avoid and put the edge-deleted graph through MinimumSpanningTree again. That feels kind of hacky, but it is easy enough to do for only ~10 MSTs if you just need them quickly.

Hey Shenghui,

HypercubeGraph returns a System`Graph, not a Combinatorica`Graph. Try this:


--William