The Structures of Zero-divisor Semigroups with Graph Kn + 1
  In this paper, we determine the structures of zero-divisor semigroups whose graph is $K_n + 1$, the complete graph $K_n$ together with an end vertex. We also present a formula to calculate the number of non-isomorphic zero-divisor semigroups corresponding to the complete graph $K_n$, for all positive integer $n$. 
Tongsuo Wu and Fan Cheng