A study on the product set-labeling of graphs

Sudev Naduvath

doi:10.1515/apam-2016-0098

Article

A study on the product set-labeling of graphs

Sudev Naduvath

Published/Copyright: January 20, 2017

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From the journal Advances in Pure and Applied Mathematics Volume 8 Issue 2

Abstract

Let X be a non-empty ground set and 𝒫⁢(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)→𝒫⁢(X) such that the induced function f∗:E⁢(G)→𝒫⁢(X) is defined by f∗⁢(u⁢v)=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)→𝒫⁢(ℕ) such that the induced edge-function f∗:V⁢(G)→𝒫⁢(ℕ) is defined as f∗⁢(u⁢v)=f⁢(u)∗f⁢(v) for all u⁢v∈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.

Keywords: Set-labeling of graphs; product set-labeling of graphs; uniform product set-labeling of graphs; geometric product set-labeling of graphs

MSC 2010: 05C78

Acknowledgements

The author would like to dedicate this work to Professor (Dr.) T. Thrivikraman, who has been his mentor, motivator and the role model in teaching as well as in research.

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Received: 2016-10-10

Revised: 2016-12-21

Accepted: 2017-1-3

Published Online: 2017-1-20

Published in Print: 2017-4-1

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Articles in the same Issue

https://doi.org/10.1515/apam-2016-0098

Keywords for this article

Set-labeling of graphs; product set-labeling of graphs; uniform product set-labeling of graphs; geometric product set-labeling of graphs