A Study on Topological Integer Additive Set-Labeling of Graphs - Sudev Naduvath (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2015)
A set-labeling of a graph G is an injective function f:V(G)→ P(X), where X is a finite set and a set-indexer of G is a set-labeling such that the induced function f^:E(G)→ P(X)-{} defined by f^(uv) = f(u)f(v) for every uv E(G) is also injective. Let G be a graph and let X be a non-empty set. A set-indexer f:V(G)→ P(X) 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(N_0), whose associated function f^+:E(G)→ P(N_0) is defined by f(uv)=f(u)+f(v), uvE(G), where N_0 is the set of all non-negative integers and P(N_0) is its power set. An integer additive set-indexer is an integer additive set-labeling such that the induced function f^+:E(G) → P(N_0) defined by f^+ (uv) = f(u)+ f(v) is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs.