am = {{0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 0, 0}, {0, 1, 0, 1, 1, 0}, {0, 
        0, 1, 1, 0, 1}, {0, 0, 1, 1, 0, 1}, {1, 0, 0, 1, 1, 1}};

    g0 = UndirectedGraph[SimpleGraph@AdjacencyGraph[am], 
      VertexCoordinates -> Reverse@CirclePoints[{1, Pi}, 6], 
      VertexLabels -> "Name"]

[![enter image description here][1]][1]

    trees = Select[TreeGraphQ[Graph@#] &] @ Select[VertexCount @ # == 6 &]@ 
      Subsets[EdgeList[g0], {5}];

    Length @ trees

>     32

This matches the result using `IGSpanningTreeCount` from `IGraphM` package:

    << IGraphM`
    IGSpanningTreeCount[g0]
>     32

    Graph[#, VertexLabels -> Placed["Name", Center], VertexStyle -> White,
        GraphLayout -> "LayeredEmbedding", 
        VertexShapeFunction -> (Disk[#, Offset[7]] &), 
        AspectRatio -> 1] & /@ trees // Multicolumn[#, 6] &

[![enter image description here][2]][2]


    HighlightGraph[g0, #, GraphHighlightStyle -> "Thick"] & /@ trees // 
     Multicolumn[#, 6, Appearance -> "Horizontal"] &

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/BnKQO.png
  [2]: https://i.sstatic.net/6BhCO.png
  [3]: https://i.sstatic.net/St0V1.png