  We consider the continuous-time quantum walk defined on the adjacency matrix
of a graph. At each instant, the walk defines a mixing matrix which is
doubly-stochastic. The average of the mixing matrices contains relevant
information about the quantum walk and about the graph. We show that it is the
matrix of transformation of the orthogonal projection onto the commutant
algebra of the adjacency matrix, restricted to diagonal matrices. Using this
formulation of the average mixing matrix, we find connections between its rank
and automorphisms of the graph.
