## Combinatorics and Graph Theory

•  Amrita Acharyya, University of Toledo
•  Co finite graphs and profi nite completions
•  11/15/2016
•  4:10 PM - 5:00 PM
•  C304 Wells Hall

Our idea of constructing a cofi nite graph starts by defining a uniform topological graph Gamma. This idea is motivated by co finite groups structure, due to B. Hartley. We define and establish a theory of co finite connectedness of a cofinite graph. We found that if G is a co finite group and Gamma = Gamma(G;X) is the Cayley graph with respect to a generating set X of G, then Gamma can be given a suitable co finite uniform topological structure so that X generates G topologically if and only if Gamma is co finitely connected. Next we develop group actions on co finite graphs. Defining the action of an abstract group over a co finite graph in the most natural way we characterize a unique way of uniformizing an abstract group with a co finite structure, obtained from the co finite structure of the graph in the underlying action, so that the aforesaid action becomes uniformly continuous. We then apply the general theory to additional structure such as groupoids, thus leading to the notions of co finite groupoids. This is a joint work with Dr. J.M.Corson (University of Alabama, Department of Mathematics) and Dr. B. Das (University of North Georgia, Department of Mathematics).

## Contact

Department of Mathematics
Michigan State University