2017/01/01 by Sudev, N. K.
#05C78 #FOS: Mathematics #General Mathematics (math.GM)
paper · doi:10.48550/arxiv.1701.00190
Let X be a non-empty ground set and \mathscrP(X) be its power set. A set-labeling (or a set-valuation) of a graph G is an injective set-valued function f:V(G)→ \mathscrP(X) such that the induced function f^*:E(G) → \mathscrP(X) is defined by f^*(uv)=f(u)∗ f(v), where f(u)∗ f(v) is a binary operation of the sets f(u) and f(v). A graph which admits a set-labeling is known to be a set-labeled graph. A set-labeling f of a graph G is said to be a set-indexer of G if the associated function f^* is also injective. In this paper, we introduce a new notion namely product set-labeling of graphs as an injective set-valued function f:V(G)→ \mathscrP(ℕ) such that the induced edge-function f^*:V(G)→ \mathscrP(ℕ) is defined as ^*f(uv)=f(u)∗ f(v) ∀ uv∈ E(G), where f(u)∗ f(v) is the product set of the set-labels f(u) and f(v), where ℕ is the set of all positive integers and discuss certain properties of the graphs which admit this type of set-labeling.