MMD Labeling of EASS of jewel graph

Abstract

Jewel graph ${\text{J}}_{\upeta } = [{\text{V}}({\text{J}}_{\upeta } ),{\text{E}}({\text{J}}_{\upeta } )]$ consists of a set of elements ${\text{V}}({\text{J}}_{\upeta } ) = \left\{ {\upalpha ,\upbeta ,\upgamma ,\updelta ,\updelta_{{\text{i}}} } \right.;1 \le {\text{i}} \le \left. \upeta \right\}$ called nodes, and another set ${\rm E}(J_{\upeta } ) = \left\{ {\upalpha \upbeta ,\upbeta \upgamma ,\upgamma \updelta ,\updelta \upalpha ,\updelta \upbeta ,\upalpha \updelta_{\text{i}} ,\upgamma \updelta_{i} } \right.;1 \le \text{i} \le \left. \upeta \right\}$, whose elements are called lines. Vertex $\updelta_{\text{i}}$ is adjacent to $\upalpha$ and $\upgamma$ such that each $\updelta_{\text{i}}$ degree is two. The prime edge in a jewel graph is defined to be the edge joining the vertices $\upbeta$&$\updelta$. 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 $\text{J}_{\upeta }$ (for $\upeta$ both odd and even values) and the EASS of the Jewel graph $\text{J}_{\upeta } ^{\prime}$ admit Modular Multiplicative Divisor labeling. Additionally, we provide related open problem.