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.

💡 Research Summary

The paper investigates the enumeration of finite topologies whose underlying undirected graph is a prescribed finite graph G. It exploits the well‑known bijections among finite topologies, preorders (reflexive and transitive relations), and directed graphs (digraphs) that arise by orienting the edges of G. A topology τ on a set X can be represented by a preorder ≤τ; each ordered pair (x, y) with x≤τ y corresponds to a directed edge x→y in a digraph Dτ. The undirected version of Dτ is exactly the given graph G, so the problem reduces to counting the number of ways to orient the edges of G such that the resulting digraph is a preorder.

The authors first formalize this reduction and then introduce group‑theoretic tools to handle isomorphism classes. The automorphism group Aut(G) acts on the set of admissible orientations, and two orientations that lie in the same orbit correspond to homeomorphic topologies. By applying Burnside’s lemma, the number of non‑homeomorphic topologies is
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