  In this paper, we study commutative zero-divisor semigroups determined by
graphs. We prove a uniqueness theorem for a class of graphs. We show two
classes of graphs that have no corresponding semigroups. In particular, any
complete graph $K_n$ together with more than three end vertices and any
complete bipartite graph together with more than one end vertices have no
corresponding semigroups. We also determine all possible zero-divisor
semigroups whose zero-divisor graph is the com- plete graph $K_3$ together with
two end vertices.
