Regular integral sum graphs

Given a set of integers S, G(S)=(S,E) is a graph, where the edge uv exists if and only if u+v∈S. A graph G=(V,E) is an integral sum graph or ISG if there exists a set S⊂Z such that G=G(S). This set is called a labeling of G. The main results of this paper concern regular ISGs. It is proved that all 2-regular graphs with the exception of C4 are integral sum graphs and that for every positive integer r there exists an r-regular ISG.