On A-Distance (Anti)magic Labeling of Prime Graph over the Ring Zn

Muhammad Husnul Khuluq

Abstract



Let G = (V, E) be a simple graph with n vertices, and let A be an abelian group of order n. Let f: V(G) → A be a bijection, and define the weight of a vertex x by w(x) = Σy ∈ N(x) f(y), where N(x) denotes the open neighborhood of x. We call f an A-distance magic labeling if all vertex weights are equal, and an A-distance antimagic labeling if the weights of distinct vertices are pairwise different. The prime graph of a commutative ring R is the simple graph with vertex set R in which two vertices x and y are adjacent if and only if xRy = {0}. In this paper, we investigate the existence of A-distance magic and A-distance antimagic labelings of the prime graph over the ring Zn for several values of n.


Keywords


Abelian group; commutative ring; A-distance antimagic; A-distance magic; prime graph.

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DOI: https://doi.org/10.18860/cauchy.v11i1.41237

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