On Topologies Induced by Graphs Under Some Unary and Binary Operations

Abstract

Let G = (V (G),E(G)) be any simple undirected graph. The open hop neighborhood of v ϵ V(G) is the set 𝑁_𝐺^2(𝑣) = {u ϵ V(G):  𝑑_𝐺 (u,v) = 2}. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G^2[A] = V(G) \ N_G^2 [A] where N_G^2 [A] = A ∪ {v ϵ V(G):  𝑁_𝐺^2(𝑣) ∩ A ≠ ∅ } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the complement of a graph, the join, the corona, the composition and the Cartesian product of graphs.