  This paper introduces a new graph construction, the permutational power of a
graph, whose adjacency matrix is obtained by the composition of a permutation
matrix with the adjacency matrix of the graph. It is shown that this
construction recovers the classical zig-zag product of graphs when the
permutation is an involution, and it is in fact more general. We start by
discussing necessary and sufficient conditions on the permutation and on the
adjacency matrix of a graph to guarantee their composition to represent an
adjacency matrix of a graph, then we focus our attention on the cases in which
the permutational power does not reduce to a zig-zag product. We show that the
cases of interest are those in which the adjacency matrix is singular. This
leads us to frame our problem in the context of equitable partitions, obtained
by identifying vertices having the same neighborhood. The families of cyclic
and complete bipartite graphs are treated in details.
