Out-graphic topology on directed graphs

Abstract

In this paper, we define a topology T_G^out, for a digraph G = (V,E) without isolated vertices called out-graphic topology, on the vertices’ set. When the graph is locally finite, T out G will be an Alexandroff topology and we give some characterisations of the minimal basis. Then, we give some open sets and some closed ones. Functions between digraphs are studied and also those between graphic topological spaces and the relations between them. Finally, for a strongly connected digraph, we prove that the topological space (V,T_G^out) can be disconnected but in other cases can be connected.