Lattice Structures in Finite Graph Topologies

Abstract

In his 2005 dissertation, Antoine Vella studied the relationships between hypergraphsand a topological space defined using graph-theoretical concepts called the classicaltopology. In this paper, we take this process a step further to open set latticesconstructed from these topological spaces. In this paper, we will characterize thestructure of these lattices entirely for finite simple graphs. We then use these resultsto conjecture on a possible relationship between hypergraphs and open set lattices.