The topological matrix of a de Bruijn diagram is quite regular,
although it is asymmetrical and differs from circulant matrices,
tridiagonal matrices, or other specialized forms which have been
extensively analyzed in the literature. De Bruijn matrices 
or
for short, are characterized by k, the number of symbols from
which sequences may be formed, and s---the number of stages---which
is the length of the sequence. In these terms, there will be k links
entering each node, as well as k links leaving each node. In turn
there will be 
nodes, with altogether 
links to join
them, corresponding to sequences of length s+1.
The uniform distribution of the numbers of incoming and outgoing links means that the topological matrix has constant row sums as well as constant column sums; therefore divided by k would be a doubly stochastic matrix with maximum eigenvalue 1 having a uniform probability distribution for its eigenvector.
It is worth presenting one or two examples and then the general case.