AN ALEXANDAN ALEXANDROFFBITOPOLOGICAL SPACE ON UNDIRECTED GRAPHSROAN ALEXANDROFFBITOPOLOGICAL SPACE ON UNDIRECTED GRAPHSFFBITOPOLOGICAL SPACE ON UNDIRECTED GRAPHS

Abstract

All nodes (vertices) has finitely many adjacent nodes(vertices) in graph G = (V, E) is calledlocally finite graph, where V is referred as vertex (node) set and E is referred as edge (arc) set.Alexandroff spaces became much more important field because of their use in digital topology.Alexandroff space is a topological space, in which arbitrary intersection of open sets is open (orarbitrary union of closed sets is closed) equivalently, we say that each singleton has minimalneighborhood base. The bitopological spaces that is the triple (A, τ1, τ2 ) of a collection A with two(arbitrary) topologies τ1and τ2on A. In this paper, we mean by a bitopologicalspace(V,τG,τIG) is anAlexandroffbitopological space, satisfy the stronger condition namely, arbitrary intersection ofmembers of SG and SIG are open in τG and τIG respectively on V, whereSG is the sub basis for agraphic topology τG and SIG is the sub basis for a incident topology τIG. Latter, we investigate someproperties and characterization of this topological spaces. In particular, the separation axioms arestudied. Our goal is to consider the fundamentalsteps toward analyzing some properties of locallyfundamental steps toward analyzing some properties of locallyfinite graphs by their corresponding topology.