Cofinite graphs and their profinite completions - Amrita Acharyya; Jon M Corson; Bikash Das (Electronic Journal of Graph Theory and Applications (EJGTA), 2017)
We generalize the idea of cofinite groups, due to B. Hartley, [2]. First we define cofinite spaces in general. Then, as a special situation, we study cofinite graphs and their uniform completions.The idea of constructing a cofinite graph starts with defining a uniform topological graph $Γ$, in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over $Γ$. It is established that for any cofinite graph there exists a unique cofinite completion.