Edge δ− graceful labeling of some snake and chain related networks
Mohamed R. Zeen El Deen, Walaa A. Aboamer, Hamed M. El-Sherbiny · Journal of Discrete Mathematical Sciences and Cryptography · 2026
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.