Orientable Group Distance Magic Labeling of Regular Graphs and Their Direct Product

Abstract

Graph labeling provides connectivity operation in networks used in computer networking, chemical structures, circuit design, and database administration.Group Distance Magic La beling (GDML) combines graph theory with group theory by using Abelian groups. A graph G has a GDML if we use elements of group for the labeling of graph’s components in such a way that the weight of each vertex in its neighborhood is contants in that group. The main focus of this work is on digraph orientable group distance magic labeling(OGDML). If an group H exits a digraph G, and if there is a injective map φ from G vertex set to the group members, then for every x ∈V, there exists a set of values such that ∑y∈N + G(x) φ→(x)− ∑y∈N − G(x) φ→(x). We study oriented graphs in this work. In particular, special labeling (OGDML) on directed graphs is the main emphasis of this study on oriented graphs.In this study, we prove that the directed direct product of Prism graphs Pn and CycleCn is OGDML under these non-isomorphic modulo groups Z2nm, Z2×Zn, Zn×Z2m, Z2×Zn/2×Z2m, and Z2 ×Z2 ×Zn/4 ×Z2m.