Enumerations of finite topologies associated with a finite graph

TL;DR

This paper focuses on enumerating finite topologies associated with a given finite graph, extending the understanding of their counts and classifications.

Contribution

It provides enumeration methods for finite topologies and non-homeomorphic topologies based on the underlying graph structure.

Findings

Enumeration formulas for topologies on specific graphs

Counts of non-homeomorphic topologies for given graphs

Connections between topologies, preorders, and digraphs

Abstract

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.