On closed distance magic circulants of valency up to 5

Abstract

Let Γ=(V,E) be a graph of order n. A closed distance magic labeling of Γ is a bijection ℓ:V→{1,2,…,n} for which there exists a positive integer r such that ∑x∈N[u]ℓ(x)=r for all vertices u∈V, where N[u] is the closed neighborhood of u. A graph is said to be closed distance magic if it admits a closed distance magic labeling.In this paper, we classify all connected closed distance magic circulants with valency at most 5, that is, Cayley graphs Cay(Zn;S) where |S|≤5, S=−S, 0∉S and S generates Zn.