Dina Khairani Nasution, Susilawati

Modular Version of The Total Vertex Irregularity Strength for The Generalized Petersen Graph

Introduction

Modular version of the total vertex irregularity strength for the generalized petersen graph. Explore the modular total vertex irregularity strength for generalized Petersen graphs. This research details a modular total irregular labeling and calculates the exact value of this graph property.

Abstract

Let  be a graph. A labeling graph is a maps function of the set of vertices and/or edges of , to the set of positive integers. A total modular labeling is said to be a -modular total irregular labeling of the vertices of , if for every two distinct vertices  and  in , the modular weights are different, and belong to the set of integers . The minimum  such that the graph  has a - modular total irregular labeling is called the modular total vertex irregularity strength and denoted by . In this paper, we study about the modular total vertex irregularity strength for the generalized Petersen graph . The result show that the exact value is .

Review

This paper introduces and investigates a novel graph invariant called the modular total vertex irregularity strength, denoted as *mts(G)*. The authors define a *k*-modular total irregular labeling as a function where distinct vertices must have unique modular weights, with these weights belonging to the set {1, ..., *k*}. The *mts(G)* is then established as the minimum possible value of *k*. The core contribution of this work is the determination of this exact value for a specific and well-known family of graphs: the generalized Petersen graph *P(n,k)*. The abstract clearly states that the main result shows the modular total vertex irregularity strength for these graphs is *3*. The study of graph labeling problems is a significant area within graph theory, often explored for its theoretical richness and potential applications. This paper proposes an interesting variant by incorporating modular arithmetic into the calculation of vertex weights, thereby creating a "modular" version of the well-established total vertex irregularity strength. This approach offers a fresh perspective on irregular labelings, which are typically concerned with ensuring distinct weights across graph elements. The choice of generalized Petersen graphs *P(n,k)* as the subject for this investigation is fitting, given their regular structure and common use as benchmarks for evaluating various graph invariants. The exact determination of *mts(P(n,k))* as *3* suggests a potentially elegant construction or a robust proof strategy demonstrating this surprisingly low minimum value. Overall, this paper presents a clear problem statement and a definitive result for a newly defined graph invariant. The introduction of the modular total vertex irregularity strength enriches the existing landscape of graph labeling theory by adding a specific arithmetic constraint, which could lead to interesting new avenues of research. The derivation of an exact value for the generalized Petersen graphs provides a foundational result for this specific invariant and graph family. This work is a valuable addition to the literature on graph irregular labelings and has the potential to inspire further investigations into the modular total vertex irregularity strength for other graph classes, as well as explorations into different types of modular labelings. The abstract effectively highlights both the problem and its significant finding.

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