Cofinite graphs and their profinite completions

Amrita Acharyya (Department of Mathematics and Statistics, University of Toledo)
Jon M Corson (Department of Mathematics, University of Alabama)
Bikash Das (Department of Mathematics, University of North Georgia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 5, No 2 (2017): Electronic Journal of Graph Theory and Applications

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 $\Gamma$, in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over $\Gamma$. It is established that for any cofinite graph there exists a unique cofinite completion.