Edge δ− graceful labeling of some snake and chain related networks

Deen, Mohamed R. Zeen El; Aboamer, Walaa A.; El-Sherbiny, Hamed M.

doi:10.47974/JDMSC-2303

Open Access Research Article

Edge δ− graceful labeling of some snake and chain related networks

^*Mohamed R. Zeen El DeenCorresponding authormohamed.zeeneldeen@sci.suezuni.edu.egDepartment of Mathematics and Computer Science Faculty of Science Suez University P. O. Box 43221Department of Mathematics and Computer Science Faculty of Science Suez UniversitySuez, 43221, EgyptView full profile → , Walaa A. Aboamerwalaa.abdelrazek@sci.suezuni.edu.egDepartment of Mathematics and Computer Science Faculty of Science Suez University P. O. Box 43221Suez, EgyptView full profile → , Hamed M. El-Sherbinyhamed.elsherbiny@sci.suezuni.edu.egDepartment of Mathematics and Computer Science Faculty of Science Suez University P. O. Box 43221Suez, EgyptView full profile →

* Corresponding author · click or hover a name for details

pp. 1–25Online FirstMarch 2026DOI: 10.47974/JDMSC-2303 Crossmark XML

Received:: 01 Aug 2024
Published Online:: 11 Mar 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2303
Pages:: 1–25

Abstract

Consider a collection of charges {δ, 2δ, 3δ, ..., βδ} for an integer δ ≥ 1 and a network Γ with α = |V (Γ)| and β = |E (Γ)|. The challenge is to create a variety of electric network designs that allow each wire in the network to employ a single charge from the set {δ, 2δ, 3δ, ..., βδ} to each wire in the network, ensuring that the amount of accumulation charges at each nodes (connection points) must vary. In this work, we developed and expanded a form of labeling of Γ entitled an edge δ– graceful labeling (Eδ – GL) which is a bijection Ψ from the edge set E(Γ) to the set {δ, 2δ, 3δ, ..., βδ} so that the produced transformation Ψ* : V(Γ) → {0, δ, 2δ, ..., (β – 1)δ}, given by: Ψ * (λ) = (∑λω∈E(Γ)‍ Ψ(λω)) mod (δΠ), where Π = max(α, β), is an injective. We provide eight designs for electric networks associated with snake graphs and four designs for electric networks associated with chain graphs. Every single one of these designs matches our required features and specifications.

Keywords

Edge δ-graceful labeling Triangular quadrilateral snake Wheel snake Linear quadrilateral chain Diagonal linear hexagonal chain

Subject Classifications

05C7805C90

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