On total edge irregularity strength of arithmetic ladder graphs

Eni Wulandari; Yeni Susanti; Eni Wulandari; Yeni Susanti

doi:10.3934/math.2026420

AIMS Mathematics

2026, Volume 11, Issue 4: 10175-10190. doi: 10.3934/math.2026420

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On total edge irregularity strength of arithmetic ladder graphs

Eni Wulandari ,
Yeni Susanti ^,

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia

Received: 25 January 2026 Revised: 22 March 2026 Accepted: 07 April 2026 Published: 14 April 2026
MSC : 05C38, 05C78

Let $ G $ be a simple, connected, and undirected graph. An edge irregular total $ \kappa $-labeling of $ G $ is a mapping that assigns each vertex and each edge an integer from $ \{1, 2, \ldots, \kappa\} $ such that distinct edges receive distinct weights, where the weight of an edge is defined as the sum of its label and the labels of its end vertices. The smallest such $ \kappa $ is called the total edge irregularity strength of $ G $, denoted by $ tes(G) $. In this paper, we determine the exact value of $ tes(G) $ for the class of arithmetic ladder graphs with $ l $ levels, $ k $ columns, and difference $ d $, where $ l\ge 2 $, $ k\ge 2 $, and $ d\ge 1 $. The results are obtained via explicit constructions of optimal total labelings.
- graph labeling,
- total edge irregularity strength,
- arithmetic ladder graph
Citation: Eni Wulandari, Yeni Susanti. On total edge irregularity strength of arithmetic ladder graphs[J]. AIMS Mathematics, 2026, 11(4): 10175-10190. doi: 10.3934/math.2026420

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Abstract

Let $ G $ be a simple, connected, and undirected graph. An edge irregular total $ \kappa $-labeling of $ G $ is a mapping that assigns each vertex and each edge an integer from $ \{1, 2, \ldots, \kappa\} $ such that distinct edges receive distinct weights, where the weight of an edge is defined as the sum of its label and the labels of its end vertices. The smallest such $ \kappa $ is called the total edge irregularity strength of $ G $, denoted by $ tes(G) $. In this paper, we determine the exact value of $ tes(G) $ for the class of arithmetic ladder graphs with $ l $ levels, $ k $ columns, and difference $ d $, where $ l\ge 2 $, $ k\ge 2 $, and $ d\ge 1 $. The results are obtained via explicit constructions of optimal total labelings.

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