Modular edge irregularity strength of graphs

Ali N. A. Koam; Ali Ahmad; Martin Bača; Andrea Semaničová-Feňovčíková; Ali N. A. Koam; Ali Ahmad; Martin Bača; Andrea Semaničová-Feňovčíková

doi:10.3934/math.2023074

AIMS Mathematics

2023, Volume 8, Issue 1: 1475-1487. doi: 10.3934/math.2023074

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Research article

Modular edge irregularity strength of graphs

1.
Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi Arabia
2.
College of Computer Science and Information Technology, Jazan University, Jazan, Saudi Arabia
3.
Department of Applied Mathematics and Informatics, Technical University, Letná 9, Košice, Slovakia
4.
Division of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, India

Received: 12 August 2022 Revised: 10 October 2022 Accepted: 17 October 2022 Published: 21 October 2022
MSC : 05C78

For a simple graph $ G = (V, E) $ with the vertex set $ V(G) $ and the edge set $ E(G) $, a vertex labeling $ \varphi: V(G) \to \{1, 2, \dots, k\} $ is called a $ k $-labeling. The weight of an edge under the vertex labeling $ \varphi $ is the sum of the labels of its end vertices and the modular edge-weight is the remainder of the division of this sum by $ |E(G)| $. A vertex $ k $-labeling is called a modular edge irregular if for every two different edges their modular edge-weights are different. The maximal integer $ k $ minimized over all modular edge irregular $ k $-labelings is called the modular edge irregularity strength of $ G $. In the paper we estimate the bounds on the modular edge irregularity strength and for caterpillar, cycle, friendship graph and $ n $-sun we determine the precise values of this parameter that prove the sharpness of the lower bound.
- (modular) irregular labeling,
- (modular) irregularity strength,
- (modular) edge irregularity strength,
- caterpillar,
- cycle,
- friendship graph,
- $ n $-sun
Citation: Ali N. A. Koam, Ali Ahmad, Martin Bača, Andrea Semaničová-Feňovčíková. Modular edge irregularity strength of graphs[J]. AIMS Mathematics, 2023, 8(1): 1475-1487. doi: 10.3934/math.2023074

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Abstract

For a simple graph $ G = (V, E) $ with the vertex set $ V(G) $ and the edge set $ E(G) $, a vertex labeling $ \varphi: V(G) \to \{1, 2, \dots, k\} $ is called a $ k $-labeling. The weight of an edge under the vertex labeling $ \varphi $ is the sum of the labels of its end vertices and the modular edge-weight is the remainder of the division of this sum by $ |E(G)| $. A vertex $ k $-labeling is called a modular edge irregular if for every two different edges their modular edge-weights are different. The maximal integer $ k $ minimized over all modular edge irregular $ k $-labelings is called the modular edge irregularity strength of $ G $. In the paper we estimate the bounds on the modular edge irregularity strength and for caterpillar, cycle, friendship graph and $ n $-sun we determine the precise values of this parameter that prove the sharpness of the lower bound.

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