Coneighbor Graphs and Related Topologies

The primary aim of this paper is to establish and analyze certain topological structures linked with a specified graph G.In a graph G, a vertex u is considered a neighbor of another vertex v if there exists an edge uv in G. Furthermore, we define two vertices (or edges) in G as coneighbors if they share identical sets of neighboring vertices (or edges).The topology under consideration arises from the collections of vertices that are coneighbor and the collections of edges that are coneighbor within the graph.It is proved that the coneighbor topology of every non-coneighbor graph is homeomorphic to the included point topology while this space is quasi-discrete if and only if the graph contains at least one coneighbor set of vertices and some examples of coneighbor topologies of special graphs are presented to be quasi-discrete spaces such as (a path, a cycle and a bipartite) graphs.Moreover, several topological properties of the coneighbor space are presented.We proved that the coneighbor topological space associated with a graph G always has dimension one and satisfies the T 1/2 axiom.Also, the family of θ-open sets is determined in this spaces and it is proved that this space is almost compact whenever the family of coneighbor sets is finite.Finally, we looked at some graphs in which the coneighbor space fulfills other topological concepts such as connectedness, compactness and countable compactness.