MMD Labeling of EASS of jewel graph

Jewel graph $${\text{J}}_{\upeta } = [{\text{V}}({\text{J}}_{\upeta } ),{\text{E}}({\text{J}}_{\upeta } )]$$ J η = [ V ( J η ) , E ( J η ) ] consists of a set of elements $${\text{V}}({\text{J}}_{\upeta } ) = \left\{ {\upalpha ,\upbeta ,\upgamma ,\updelta ,\updelta_{{\text{i}}} } \right.;1 \le {\text{i}} \le \left. \upeta \right\}$$ V ( J η ) = α , β , γ , δ , δ i ; 1 ≤ i ≤ η called nodes, and another set $${\rm E}(J_{\upeta } ) = \left\{ {\upalpha \upbeta ,\upbeta \upgamma ,\upgamma \updelta ,\updelta \upalpha ,\updelta \upbeta ,\upalpha \updelta_{\text{i}} ,\upgamma \updelta_{i} } \right.;1 \le \text{i} \le \left. \upeta \right\}$$ E ( J η ) = α β , β γ , γ δ , δ α , δ β , α δ i , γ δ i ; 1 ≤ i ≤ η , whose elements are called lines. Vertex $$\updelta_{\text{i}}$$ δ i is adjacent to $$\upalpha$$ α and $$\upgamma$$ γ such that each $$\updelta_{\text{i}}$$ δ i degree is two. The prime edge in a jewel graph is defined to be the edge joining the vertices $$\upbeta$$ β & $$\updelta$$ δ . A graph L (V, E) with |V|= n is said to have modular multiplicative divisor labeling if there exists a bijection f: V(L) → {1, 2, …,n} and the induced function f*: E(L) → {0, 1, 2, …, n − 1} where f*(uv) = f(u)f(v) (mod n) such that n divides the sum of all edge labels of L. In this paper, we prove that both the Jewel graph $$\text{J}_{\upeta }$$ J η (for $$\upeta$$ η both odd and even values) and the EASS of the Jewel graph $$\text{J}_{\upeta } ^{\prime}$$ J η ′ admit Modular Multiplicative Divisor labeling. Additionally, we provide related open problem.