The Non-Adjacency Incompatible Vertices Topology on Digraphs

In this work, we introduce a new type of topology associated to directed graph on vertices set, which is named as non-adjacency incompatible vertices topology (𝑁𝐼𝑉-topology) of directed graph. A sub-basis family of this topology is generated on the set of vertices, and formed via taking the non-adjacency vertices which arises (path of length two) in different direction to each vertex. We investigate some properties and discuss it on an important and certain types of directed graphs. Our motivation is to giving fundamental steps toward investigation of various properties of directed graphs via their corresponding non-adjacency incompatible vertices topology. Some results are obtained, the presence of the isolated vertex is not necessary in our definition, observations about the specification of this new topology with certain types of digraphs (which type of digraphs achieves the discrete non-adjacency incompatible vertices topology and which does not, and when it achieves 𝑇1and 𝑇2-space). Furthermore, application (applied example in biomathematics) to the Pulmonary circulation in humans is introduced in this paper.