Modular Multiplicative Divisor Labeling of Some Path Related Graphs

R Revathi

doi:10.17485/ijst/2014/v7i12.25

Abstract

A graph G (V,E) with |V| = n is said to have modular multiplicative divisor labeling if there exist bijection f: V(G) → {1, 2, …,} and the induced function f * : E(G) → {0, 1, 2, …, n - 1} where f(uv) = f(u)f(v) (mod n) such that n divides the sum of all edge labels of G. We prove that the path P n , and the graph P a, (a graph which connects two vertices by means of b internally disjoint paths of length a each), shadow graph of path and the cartesian product P n × P 1 , (n is not multiple of 6) admits modular multiplicative divisor labeling. Also we discuss the upper bound for the number of edges in modular multiplicative divisor graphs. AMS Subject Classification: 05C78

Highlights

For all terminology and notation in graph theory we follow Harary[4]
We prove that the path Pn, and the graph Pa, b, shadow graph of a path and the cartesian product Pn × P1, (n is not a multiple of 6) admits modular multiplicative divisor labeling
We discuss the upper bound for the number of edges in a modular multiplicative divisor graphs

Summary

Introduction

For all terminology and notation in graph theory we follow Harary[4]. In this paper we consider only finite, simple, connected and undirected graphs. Modular multiplication plays an important role in number theory problems[2]. Through some mathematical logic we could able to introduce a new labeling called modular multiplicative divisor (MMD) labeling and we proved[3] complete graph Kn, for all prime number n > 3, complete bipartite graph Km,n , cycle graph Cn, n ≡ 1, 2(mod3) are modular multiplicative divisor graphs. We proved[9] split graph of cycle Cn, helm graph Hn, flower graph fn × 4, cycle cactus C4(n), extended triplicate graph of a path are modular multiplicative divisor graphs. Modular multiplicative divisor labeling techniques can be applied in the field of cryptography[7]. The present work is intended to discuss the existence of MMD labeling of a path and path related graphs and the maximum number of edges in a modular multiplicative divisor graphs

Definitions Related to the Context

Discussion of MMD Labeling of Path Related Graphs

V3 4 V4

Upper Bound for Number of Edges of a Modular Multiplicative Divisor Graphs

Conclusion