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Given a topological space, the graphical realizations of it with as many edges as possible, called maximal graphical realizations, are studied here. Every finite topological space admits a maximal graphical realization. However, there are graphs which are not maximal graphical realizations of any topology. A tree of odd order is never a maximal graphical realization of a topological space. Maximal graphical realization of a topology is a cycle if and only if it is C₃. It is shown that chain topologies admit unique maximal graphical realizations. A lower bound for the size of a maximal graphical realization is also obtained.
Thomas et al. (2024) studied this question.