Enumerations of finite topologies associated with a finite graph
  The number of topologies and non-homeomorphic topologies on a fixed finite set are now known up to $n=18$, $n=16$ but still no complete formula yet (Sloane). There are one to one correspondence among topologies, preorder and digraphs. In this article, we enumerate topologies and non-homeomorphic topologies whose underlying graph is a given finite graph. 
Dongseok Kim, Young Soo Kwon and Jaeun Lee
math.CO math.GN