  Quantum discord refers to an important aspect of quantum correlations for
bipartite quantum systems. In our earlier works we have shown that
corresponding to every graph (combinatorial) there are quantum states whose
properties are reflected in the structure of the corresponding graph. Here, we
attempt to develop a graph theoretic study of quantum discord that corresponds
to a necessary and sufficient condition of zero quantum discord states which
says that the blocks of density matrix corresponding to a zero quantum discord
state are normal and commute with each other. These blocks have a one to one
correspondence with some specific subgraphs of the graph which represents the
quantum state. We obtain a number of graph theoretic properties representing
normality and commutativity of a set of matrices which are indeed arising from
the given graph. Utilizing these properties we define graph theoretic measures
for normality and commutativity that results a formulation of graph theoretic
quantum discord. We identify classes of quantum states with zero discord using
the said formulation.
