  This work is at the interface of graph theory and quantum mechanics. Quantum
correlations epitomize the usefulness of quantum mechanics. Quantum discord is
an interesting facet of bipartite quantum correlations. Earlier, it was shown
that every combinatorial graph corresponds to quantum states whose
characteristics are reflected in the structure of the underlined graph. A
number of combinatorial relations between quantum discord and simple graphs
were studied. To extend the scope of these studies, we need to generalize the
earlier concepts applicable to simple graphs to weighted graphs, corresponding
to a diverse class of quantum states. To this effect, we determine the class of
quantum states whose density matrix representation can be derived from graph
Laplacian matrices associated with a weighted directed graph and call them
graph Laplacian quantum states. We find the graph-theoretic conditions for zero
and non-zero quantum discord for these states. We apply these results on some
important pure two qubit states, as well as a number of mixed quantum states,
such as the Werner, Isotropic, and $X$-states. We also consider graph Laplacian
states corresponding to simple graphs as a special case.
