Modular Irregular Labeling on Double-Star and Friendship Graphs

K A Sugeng,Z Z Barack,N Hinding,R Simanjuntak,Ali Jaballah

doi:10.1155/2021/4746609

K A Sugeng, Z Z Barack + Show 3 more

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https://doi.org/10.1155/2021/4746609

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Abstract

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

Highlights

Graph labeling is a mapping of a set of numbers, called the labels, to the graph elements, usually vertices or edges [1]
The label is a positive integer. ere are several labelings that have been developed; among them are irregular labeling and modular irregular labeling. e reader can check the dynamic survey of graph labeling by Gallian to obtain more information on various labeling [1]
An irregular labeling is defined as a labeling f: E ⟶ {1, 2, . . . , k} with k as a positive integer, such that wtf(x) 􏽐(y∈N(x))f(xy) is different for all vertices, where N(x) is a neighbour of vertex x. e irregularity strength s(G) of a graph G is the minimum value of k for which G has irregular labeling with labels at most k. e irregularity strength s(G) of a graph G is defined only for graphs containing at most one isolated vertex and no connected component of order 2. e lower bound of the irregularity strength of a graph G is s(G) ≥ max1≤i≤△ 􏼈ni + i − 1/i􏼉, where ni vertices with degree i, as stated in eorem 1

Summary

Introduction

Graph labeling is a mapping of a set of numbers, called the labels, to the graph elements, usually vertices or edges [1]. Modular irregular labeling of a graph is a mapping φ: E(G) ⟶ {1, 2, . Baca et al [10] proved the modular irregularity strength of the fan graph.

Results

Conclusion