On topological spaces associated with digraphs

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

“…So, by Theorem 4.3, e / ∈ intMP (E(D) \ A); hence, e ∈ E(D) \ intMP (E(D) \ A). By Proposition 2.7 (6), e ∈ clMP (A). Therefore, A ∪ A ⊆ clMP (A).…”

“…In 1968, Bhargava T. N. and Ahlborn T. J. [6] showed that each directed graph D = (V, E) defines a unique topological space (V, τE) where τE = {U : U ∈ 2 V , U open}; and the subset U is said to define an open set if for every pair of points (u, v) with v in U and u not in U that is (u, v) / ∈ E. In 1972, Lieberman R. N. [7] defined two topologies on the set of vertices of every directed graph D = (V, E) called the left E-topology and the right E-topology. He illustrated that the left E-topology is equivalent to the topology presented by Bhargava T. N. and Ahlborn T. J.…”

“…He illustrated that the left E-topology is equivalent to the topology presented by Bhargava T. N. and Ahlborn T. J. [6]. In 2010, Marijuan C. [8] studied the relation between directed graphs and finite topologies for which he associated a topology τ with the vertex set of each directed graph.…”

Topologies on the edges set of directed graphs

“…The left E-topology is the collection of all subsets of such that if ∈ and ( , ) ∈ then ∈ while the right E-topology is the collection of all subsets of such that if ∈ and ( , ) ∈ then ∈ . He illustrated that the left E-topology is equivalent with the topology presented by T. N. Bhargava and T. J. Ahlborn [3]. In 1973, E. Sampathkumar and K. H. Kulkarni [12] generalized the results provided by J. W. Evans et al [8].…”

“…In 1967, S. S. Anderson and G. Chartrand [1] investigate the lattice-graph of the topologies of transitive directed graphs presented by J. W. Evans et al [8]. In 1968, T. N. Bhargava and T. J. Ahlborn [3] shows that each directed graph = ( , ) defines a unique topological apace ( , ) where = { : ∈ 2 , open}; and the subset is said to define an open set if for every pair of points ( , ) with in and not in , ( , ) ∉ . In 1972, R. N. Lieberman [9] defines two topologies on the set of nodes of every directed graph = ( , ), called the left E-topology and the right E-topology.…”