On magic distinct labellings of simple graphs

A magic labelling of a graph G with sum s is the edges by nonnegative integers such that for each vertex v∈V, labels all incident to v equal same number s. Stanley gave remarkable results on labellings, but distinct case much more complicated. We consider complete construction labellings given G. The idea illustrated in detail dealing three regular graphs and non-regular graph. give combinatorial proofs. structure result was used enumerate corresponding labellings. can be carried out directly no than 9 edges.