  Representing graphs as quantum states is becoming an increasingly important
approach to study entanglement of mixed states, alternate to the standard
linear algebraic density matrix-based approach of study. In this paper, we
propose a general weighted directed graph framework for investigating
properties of a large class of quantum states which are defined by three types
of Laplacian matrices associated with such graphs. We generalize the standard
framework of defining density matrices from simple connected graphs to density
matrices using both combinatorial and signless Laplacian matrices associated
with weighted directed graphs with complex edge weights and with/without
self-loops. We also introduce a new notion of Laplacian matrix, which we call
signed Laplacian matrix associated with such graphs. We produce necessary
and/or sufficient conditions for such graphs to correspond to pure and mixed
quantum states. Using these criteria, we finally determine the graphs whose
corresponding density matrices represent entangled pure states which are well
known and important for quantum computation applications. We observe that all
these entangled pure states share a common combinatorial structure.
