Reflexive edge strength of convex polytopes and corona product of cycle with path

Kooi-Kuan Yoong; Roslan Hasni; Gee-Choon Lau; Muhammad Ahsan Asim; Ali Ahmad; Kooi-Kuan Yoong; Roslan Hasni; Gee-Choon Lau; Muhammad Ahsan Asim; Ali Ahmad

doi:10.3934/math.2022657

AIMS Mathematics

2022, Volume 7, Issue 7: 11784-11800. doi: 10.3934/math.2022657

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Research article

Reflexive edge strength of convex polytopes and corona product of cycle with path

1.
Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Terengganu, Malaysia
2.
Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (Segamat Campus), Johor, Malaysia
3.
College of Computer Sciences and Information Technology, Jazan University, Jazan, Saudi Arabia

Received: 25 December 2021 Revised: 16 February 2022 Accepted: 21 February 2022 Published: 18 April 2022
MSC : 05C78, 05C38

For a graph $ G $, we define a total $ k $-labeling $ \varphi $ is a combination of an edge labeling $ \varphi_e(x)\to\{1, 2, \ldots, k_e\} $ and a vertex labeling $ \varphi_v(x) \to \{0, 2, \ldots, 2k_v\} $, such that $ \varphi(x) = \varphi_v(x) $ if $ x\in V(G) $ and $ \varphi(x) = \varphi_e(x) $ if $ x\in E(G) $, then $ k = \, \mbox{max}\, \{k_e, 2k_v\} $. The total $ k $-labeling $ \varphi $ is an edge irregular reflexive $ k $-labeling of $ G $ if every two different edges $ xy $ and $ x^\prime y^\prime $, the edge weights are distinct. The smallest value $ k $ for which such labeling exists is called a reflexive edge strength of $ G $. In this paper, we focus on the edge irregular reflexive labeling of antiprism, convex polytopes $ \mathcal D_{n} $, $ \mathcal R_{n} $, and corona product of cycle with path. This study also leads to interesting open problems for further extension of the work.
- convex polytope,
- corona product,
- edge irregular reflexive labeling,
- plane graph,
- reflexive edge strength
Citation: Kooi-Kuan Yoong, Roslan Hasni, Gee-Choon Lau, Muhammad Ahsan Asim, Ali Ahmad. Reflexive edge strength of convex polytopes and corona product of cycle with path[J]. AIMS Mathematics, 2022, 7(7): 11784-11800. doi: 10.3934/math.2022657

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Abstract

For a graph $ G $, we define a total $ k $-labeling $ \varphi $ is a combination of an edge labeling $ \varphi_e(x)\to\{1, 2, \ldots, k_e\} $ and a vertex labeling $ \varphi_v(x) \to \{0, 2, \ldots, 2k_v\} $, such that $ \varphi(x) = \varphi_v(x) $ if $ x\in V(G) $ and $ \varphi(x) = \varphi_e(x) $ if $ x\in E(G) $, then $ k = \, \mbox{max}\, \{k_e, 2k_v\} $. The total $ k $-labeling $ \varphi $ is an edge irregular reflexive $ k $-labeling of $ G $ if every two different edges $ xy $ and $ x^\prime y^\prime $, the edge weights are distinct. The smallest value $ k $ for which such labeling exists is called a reflexive edge strength of $ G $. In this paper, we focus on the edge irregular reflexive labeling of antiprism, convex polytopes $ \mathcal D_{n} $, $ \mathcal R_{n} $, and corona product of cycle with path. This study also leads to interesting open problems for further extension of the work.

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