Distance magic circulant graphs

Let G=(V,E)G=(V,E) be a graph of order nn. A distance magic labeling of GG is a bijection ℓ:V→{1,2,…,n}ℓ:V→{1,2,…,n} for which there exists a positive integer kk such that ∑x∈N(v)ℓ(x)=k∑x∈N(v)ℓ(x)=k for all v∈Vv∈V, where N(v)N(v) is the neighborhood of vv. In this paper we deal with circulant graphs Cn(1,p)Cn(1,p). The circulant graph Cn(1,p)Cn(1,p) is the graph on the vertex set V={x0,x1,…,xn−1}V={x0,x1,…,xn−1} with edges (xi,xi+p)(xi,xi+p) for i=0,…,n−1i=0,…,n−1 where i+pi+p is taken modulo nn. We completely characterize distance magic graphs Cn(1,p)Cn(1,p) for pp odd. We also give some sufficient conditions for pp even. Moreover, we also consider a group distance magic labeling of Cn(1,p)Cn(1,p).