  We have generalised the concept of graph states to what we have called mixed
graph states, which we define in terms of mixed graphs, that is graphs with
both directed and undirected edges, as the density matrix stabilized by the
associated stabilizer matrix defined by the mixed graph. We can interpret this
matrix as a quantum object by making it part of a larger fully commuting
matrix, i.e. where we choose the environment appropriately, and this will imply
that our quantum object is a mixed state.
  We prove that, in the same way as (pure) graph states, the density matrix of
a parent of mixed graph state can be written as sum of a few Pauli matrices,
well defined from the mixed graph. We have proven that the set of matrices that
appear in this sum is fully pair-wise commuting, and form a multiplicative
group up to global constants, which is always of maximum size. Furthermore, the
cardinality of the set depends solely of the miminum possible number of
extension columns/rows, and the number of nodes of the mixed graph. We prove a
formula for this cardinality. Finally, in the case of purely undirected graphs,
this corresponds to the usual pure graph state.
  Also, we have developed a way of finding maximal commutative group of such
Pauli matrices as a linear subspace problem, for any given mixed graph. We also
have proven how the structure of maximal commutative groups is independent of
the direction of the arrows of the mixed graph, and also of the undirected
edges; this allows the simplification of the problem of finding these groups in
general to finding them for a much smaller set of graphs.
