Gm-Closure Spaces and Related Results

The purpose of the present work is to put a starting point for the applications of abstract topological graph theory into rough set theory, granular computing and fuzzy set theory. Also, we will integrate some ideas in terms of concepts in topological graph theory and uncertainty. Topology and its applications occupy the interest of many researching centers in the advanced world. From this point of view and because the near open sets play a very important role in general topology and they are now the research topics of many topologists worldwide and there is a very important role of closure operators in constructing a topological spaces, so we introduce a new closure operators on the power set of vertices on graphs and conclusion theorems and new spaces from it. Furthermore, we discuss the relationships of connectedness between some types of graphs and new spaces by using graph closure operators. The boundary regions in approximation spaces are considered as uncertainty regions. We also offer some theories of uncertainty through the topological spaces which result from new closure operator of graphs on the approximation spaces. Finally, we study some related applications.