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The number of spanning trees of a graph $G$ is called the {\em complexity} of$G$ and is denoted $c(G)$. Let C(n) denote the {\em (binary) hypercube} ofdimension $n$. A classical result in enumerative combinatorics (based onexplicit diagonalization) states that $c(C(n)) = \prod_{k=2}^n (2k)^{n\choosek}$.  In this paper we use the explicit block diagonalization methodology to deriveformulas for the complexity of two $q$-analogs of C(n), the {\em nonbinaryhypercube} $\Cq(n)$, defined for $q\geq 2$, and the {\em vector space analog ofthe hypercube} $\Cfq(n)$, defined for prime powers $q$.  We consider the nonbinary and vector space analogs of the Boolean algebra. Weshow the existence, in both cases, of a graded Jordan basis (with respect tothe up operator) that is orthogonal (with respect to the standard innerproduct) and we write down explicit formulas for the ratio of the lengths ofthe successive vectors in the Jordan chains (i.e., the singular values). Withrespect to (the normalizations of) these bases the Laplacians of $\Cq(n)$ and$\Cfq(n)$ block diagonalize, with quadratically many distinct blocks in thenonbinary case and linearly many distinct blocks in the vector space case, andwith each block an explicitly written down real, symmetric, tridiagonal matrixof known multiplicity and size at most $n+1$. In the nonbinary case we furtherdetermine the eigenvalues of the blocks, by explicitly writing out theeigenvectors, yielding an explicit formula for $c(\Cq(n))$ (this proof yieldsnew information even in the binary case). In the vector space case we have beenunable to determine the eigenvalues of the blocks but we give a useful formulafor $c(\Cfq(n))$ involving "small" determinants (of size at most $n$).

Counting spanning trees of the hypercube and its $q$-analogs by explicit block diagonalization