A Generalized Topology from the Edge Set of Maximal Paths of Directed Graphs

This study intends to provide a fundamental step towards studying the properties of directedgraphs with their corresponding generalized topological spaces. A generalized topology (GT)μon a nonempty setXis de ned as a family of subsets ofXsuch that∅and an arbitrary unionof sets inμis inμ. In this study, we introduce a new generalized topology generated by the setof edges of maximal paths of the directed graphDcalled the maximal path edge generalizedtopology (MPE), denoted by MP(D). The basic topological properties and connectednessin the context of this new structure are explored and illustrated. In particular, this paperestablished that (E(D),MP(D)) is a strong generalized topological space and characterizedthe open and closed sets in this space. Moreover, it was seen that the MPE space of everydisconnected digraph is MP-disconnected and the MPE space of every connected digraph isalso characterized.