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 in a formulation of graph theoretic quantum discord. We identify classes of quantum states with zero discord using the developed formulation. © 2017, Springer Science+Business Media, LLC.