Group Distance Magic Labeling On Some Classes Of Cayley Graphs

Let G=(V,E)and G=(V,E)be simple undirected and directed graphs with n vertices.Γ is a finite Abelian group of order n.A Γ-distance magic labeling of G is a bijection φ:V(G)→Γ with the property that there exists c∈Γ such that ∑x∈N(v)φ(x)=c for any v∈V(G),where N(v)is the neighborhood of x.The constant c is called the magic constant of the magic labeling.A graph G is called Γ-distance magic if it admits a Γ-distance magic labeling.The digraph is denoted by G.A Γ-distance magic labeling of G is a bijection φ:V(G)→Γ with the property that there exists a pastive integer k such that ∑y∈N+(x)φ(y)-∑y∈N-(x)φ(y)=k for any x ∈V(G).The constant k is called the magic constant of this labeling.Where N+(x)={y ∈V(G):yx ∈ E(G)} and N-(x)={y∈ V(G):xy ∈E(G)｝.The framework of this paper is as follows:In Chapter 1 we first introduce the research background and development history of distance magic labeling.In this paper,We introduce common notations and terminology in distance magic labeling,and some properties about the period of the sequence of finite abelian groups are required.In Chapter 2 we determine exactly on which abelian groups there exist group distance magic labeling for square power graphs of cycles.In Chapter 3 we determine exactly on which abelian groups there exist group distance magic labeling for Cartesian products of two cycles..At the same time,we have also fully determined the possible values of the magic constant when the Cartesian product has an Zmn-distance magic labeling.In Chapter 4 we determine exactly on which abelian groups there exist group distance magic labeling for Cartesian products of two directed cycles.