A new generalization of edge-irregular evaluations

Martin Bača; Muhammad Imran; Zuzana Kimáková; Andrea Semaničová-Feňovčíková; Martin Bača; Muhammad Imran; Zuzana Kimáková; Andrea Semaničová-Feňovčíková

doi:10.3934/math.20231287

AIMS Mathematics

2023, Volume 8, Issue 10: 25249-25261. doi: 10.3934/math.20231287

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Research article Special Issues

A new generalization of edge-irregular evaluations

1.
Department of Applied Mathematics and Informatics, Technical University, Letná 9, Košice, Slovakia
2.
Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates
3.
Division of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, India

Received: 18 April 2023 Revised: 14 August 2023 Accepted: 18 August 2023 Published: 30 August 2023
MSC : 05C78

Consider a simple graph $ G = (V, E) $ of size $ m $ with the vertex set $ V $ and the edge set $ E $. A modular edge-irregular total $ k $-labeling of a graph $ G $ is a labeling scheme for the vertices and edges with the labels $ 1, 2, \dots, k $ that allows the modular weights of any two different edges to be distinct, where the modular weight of an edge is the remainder of the division of the weight (i.e., the sum of the label of the edge itself and the labels of its two end vertices) by $ m $. The maximal integer $ k $, minimized over all modular edge-irregular total $ k $-labelings of the graph $ G $ is called the modular total edge-irregularity strength. In the paper, we generalize the approach to edge-irregular evaluations, introduce the notion of the modular total edge-irregularity strength and obtain its boundary estimation. For certain families of graphs, we investigate the existence of modular edge-irregular total labelings and determine the precise values of the modular total edge-irregularity strength in order to prove the sharpness of the lower bound.
- modular edge-irregular labeling,
- modular edge-irregularity strength,
- modular total edge-irregularity strength,
- cycle,
- $ n $-sun,
- circulant graph
Citation: Martin Bača, Muhammad Imran, Zuzana Kimáková, Andrea Semaničová-Feňovčíková. A new generalization of edge-irregular evaluations[J]. AIMS Mathematics, 2023, 8(10): 25249-25261. doi: 10.3934/math.20231287

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Abstract

Consider a simple graph $ G = (V, E) $ of size $ m $ with the vertex set $ V $ and the edge set $ E $. A modular edge-irregular total $ k $-labeling of a graph $ G $ is a labeling scheme for the vertices and edges with the labels $ 1, 2, \dots, k $ that allows the modular weights of any two different edges to be distinct, where the modular weight of an edge is the remainder of the division of the weight (i.e., the sum of the label of the edge itself and the labels of its two end vertices) by $ m $. The maximal integer $ k $, minimized over all modular edge-irregular total $ k $-labelings of the graph $ G $ is called the modular total edge-irregularity strength. In the paper, we generalize the approach to edge-irregular evaluations, introduce the notion of the modular total edge-irregularity strength and obtain its boundary estimation. For certain families of graphs, we investigate the existence of modular edge-irregular total labelings and determine the precise values of the modular total edge-irregularity strength in order to prove the sharpness of the lower bound.

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