A Study on Topological Integer Additive Set- Labeling of Graphs

A set-labeling of a graph G is an injective function f : V (G)→P(X) such that the induced function f(circled plus) : E(G)→P(X)-(phi) f;g defined by f(circled plus)(uv) = f(u)(circled plus)f(v) for every uv(element of)E(G), where X is a non-empty finite set and P(X) be its power set. A set-indexer of G is a set-labeling such that the induced function f(circled plus) is also injective. A set-indexer f : V (G)→P(X) of a given graph G is called a topological set-labeling of G if f(V (G)) is a topology of X. An integer additive set-labeling is an injective function f : V (G)→P(N0), whose associated function f+ : E(G)→P(N0) is defined by f+(uv) = f(u) + f(v); uv 2 E(G), where N0 is the set of all non-negative integers. An integer additive set-indexer is an integer additive set-labeling such that the induced function f+ : E(G)→P(N0) is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs.