On Topologies Induced by Graphs Under Some Unary and Binary Operations Caen

Let G = (V (G), E(G)) be any simple undirected graph. The open hop neighborhood of v ∈ V (G) is the set N G(v) = {u ∈ V (G) : dG(u, v) = 2}. Then G induces a topology τG on V (G) with base consisting of sets of the form F 2 G[A] = V (G)\N G[A], where N G[A] = A ∪ {v ∈ V (G) : N G(v) ∩ A 6= ∅} 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. 2010 Mathematics Subject Classifications: 05C76


Introduction
Let G = (V (G), V (H)) be any simple undirected graph.The distance d(u, v) between two vertices u and v in G is the length of a shortest path joining u and v. Let v ∈ V (G).The neighborhood of v is the set N (v) consisting of all u ∈ V (G) which are adjacent with v and the closed neighborhood is N [v] = N (v) ∪ {v}.For any A ⊆ V (G), N (A) = {x : xa ∈ E(G) for some a ∈ A} is called the neighborhood of A and In 1983, Diesto and Gervacio in [5] proved that given a simple graph G = (V (G), E(G)), G induces a topology on V (G), denoted by τ G , with base consisting of sets of the form , where N G (A) = A ∪ {x : xa ∈ E for some a ∈ A} and A ranges over all subsets of V (G).Their construction was further investigated in [2], [3] and [6].In particular, Canoy and Lemence in [2] described the topologies induced by the complement of a graph, the join of graphs, composition and Cartesian product of graphs.
In [1], Canoy and Gimeno presented another way of constructing a topology τ G from a connected graph G by considering the family Ω = 2 for all a ∈ A}.They showed that this family is a base for some topology τ G on V (G).This construction is also studied by Nianga et al, in [4] for any graph G.It is also shown that the family } are, respectively, base and subbase for the topology τ G on V (G).
Concepts on Graph Theory and Topology are taken from [7] and [8], respectively.

Results
Definition 1.The complement of graph G, denoted by G is the graph with Theorem 1.Let G be any graph and G its complement.Then for each v ∈ V (G), Proof.Let G be any graph and G its complement.Let v ∈ V (G) and set . This establishes the desired equality.
Theorem 2. Let G be any graph and Remark 1.The converse of theorem 17 is not true. Hence, Remark 2. Let G be any graph and let ) and H = (V (H), E(H)) be graphs.Then for any The next theorem follows from Theorem 3 (i) and (ii).
Definition 3. The corona G • H of graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies H and then forming the sum v , where H v is a copy of H corresponding to the vertex v.
Definition 4. The lexicographic product (composition) of graphs G and H, denoted by ) and H = (V (H), E(H)) be any two graphs and let . Hence, t = a and d H (a, t) = 2. Consequently, (z, t) = (v, a) and a)].This establishes the desired equality.