Modular total vertex irregularity strength of graphs

Gohar Ali; Martin Bača; Marcela Lascsáková; Andrea Semaničová-Feňovčíková; Ahmad ALoqaily; Nabil Mlaiki; Gohar Ali; Martin Bača; Marcela Lascsáková; Andrea Semaničová-Feňovčíková; Ahmad ALoqaily; Nabil Mlaiki

doi:10.3934/math.2023384

AIMS Mathematics

2023, Volume 8, Issue 4: 7662-7671. doi: 10.3934/math.2023384

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Research article

Modular total vertex irregularity strength of graphs

1.
Department of Mathematics, Islamia College Peshawar, Khyber Pakhtunkhwa, Pakistan
2.
Department of Applied Mathematics and Informatics, Technical University, Letná 9, Košice, Slovakia
3.
Division of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, India
4.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
5.
School of computer, Data and Mathematical Sciences, Western Sydney University, Sydney, Australia

Received: 16 November 2022 Revised: 10 January 2023 Accepted: 12 January 2023 Published: 18 January 2023
MSC : 05C78

A (modular) vertex irregular total labeling of a graph $ G $ of order $ n $ is an assignment of positive integers from $ 1 $ to $ k $ to the vertices and edges of $ G $ with the property that all vertex weights are distinct. The vertex weight of a vertex $ v $ is defined as the sum of numbers assigned to the vertex $ v $ itself and to the edge's incident, while the modular vertex weight is defined as the remainder of the division of the vertex weight by $ n $. The (modular) total vertex irregularity strength of $ G $ is the minimum $ k $ for which such labeling exists. In this paper, we obtain estimations on the modular total vertex irregularity strength, and we evaluate the precise values of this invariant for certain graphs.
- (modular) irregular labeling,
- (modular) irregularity strength,
- (modular) total vertex irregularity strength
Citation: Gohar Ali, Martin Bača, Marcela Lascsáková, Andrea Semaničová-Feňovčíková, Ahmad ALoqaily, Nabil Mlaiki. Modular total vertex irregularity strength of graphs[J]. AIMS Mathematics, 2023, 8(4): 7662-7671. doi: 10.3934/math.2023384

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Abstract

A (modular) vertex irregular total labeling of a graph $ G $ of order $ n $ is an assignment of positive integers from $ 1 $ to $ k $ to the vertices and edges of $ G $ with the property that all vertex weights are distinct. The vertex weight of a vertex $ v $ is defined as the sum of numbers assigned to the vertex $ v $ itself and to the edge's incident, while the modular vertex weight is defined as the remainder of the division of the vertex weight by $ n $. The (modular) total vertex irregularity strength of $ G $ is the minimum $ k $ for which such labeling exists. In this paper, we obtain estimations on the modular total vertex irregularity strength, and we evaluate the precise values of this invariant for certain graphs.

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