How to find the number of spanning trees for a cube? Can you tell me a way of finding the total number of spanning trees in a $Q_d$ undirected labelled graph for $d = 3$. 
I know that the answer is 384, but the way (I know there are many.) of finding it escapes me. 

One of the most interesting theorems about the number of spanning trees of a graph is matrix tree theorem.
Matrix Tree Theorem states that: Given a loopless graph G with vertex set $v_1,v_2,...,v_s$, let D be a diagonal matrix in which $d_{ii}=deg_G(v_i)$, and let A be the adjacency matrix of G. Define $Q:=D-A$. If $Q^*$ is a matrix obtained by deleting row $s$ and column $t$ of $ Q$, then the number of spanning trees of G is equal to $(-1)^{t+s}det(Q^*)$.
You can find the proof here:
Introduction to Graph Theory / Douglas B. West / page 86

For your question, $Q^*$ will be a $7 \times 7$ matrix, but it still isn't hard to calculate it's determinant.
