  In this paper we consider the separability problem for bipartite quantum
states arising from graphs. Earlier it was proved that the degree criterion is
the graph-theoretic counterpart of the familiar positive partial transpose
criterion for separability, although there are entangled states with positive
partial transpose for which the degree criterion fails. Here we introduce the
concept of partially symmetric graphs and degree symmetric graphs by using the
well-known concept of partial transposition of a graph and degree criteria,
respectively. Thus, we provide classes of bipartite separable states of
dimension $m \times n$ arising from partially symmetric graphs. We identify
partially asymmetric graphs that lack the property of partial symmetry. We
develop a combinatorial procedure to create a partially asymmetric graph from a
given partially symmetric graph. We show that this combinatorial operation can
act as an entanglement generator for mixed states arising from partially
symmetric graphs.
