In his 2005 dissertation, Antoine Vella explored combinatorical aspects of finite graphs utilizing a topological space whose open sets are intimately tied to the structure of the graph. In this paper, we go a step further and examine some aspects of the open set lattices induced by these topological spaces. In particular, we will characterize all lattices that constitute the opens for finite simple graphs endowed with this topology, explore the structure of these lattices, and show that these lattices contain information necessary to reconstruct the graph and its complement in several ways.
Hart, James B. and Frazier, Brian
"Finite Simple Graphs and Their Associated Graph Lattices,"
Theory and Applications of Graphs: Vol. 5
, Article 6.
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol5/iss2/6