  For any graph, we define a rank-1 operator on a bipartite tensor product
space, with components associated to the set of vertices and edges
respectively. We show that the partial traces of the operator are the Laplacian
and the edge-Laplacian. This provides an interpretation of the von Neumann
entropy of the (normalized)\ Laplacian as the amount of quantum entanglement
between two systems corresponding to vertices and edges. In this framework,
cospectral graphs correspond exactly to local unitarily equivalent pure states.
Finally, we introduce the notion of coentropic graphs, that is, graphs with
equal von Neumann entropy. The smallest coentropic (but not cospectral) graphs
that we are able to construct have 8 vertices. The number of equivalence
classes of coentropic graphs with n vertices and m edges is a lower bound to
the number of (pure) bipartite entanglement classes with subsystems of
corresponding dimension.
