On the sum number of wheels

A graph G=(V,E) is a sum graph if there exists a labelling of the vertices of G with distinct positive integers so that uv∈E if and only if the sum of the labels assigned to u and v is the label of a vertex of G. The sum number of a graph is the minimum number of isolated vertices that, together with the given graph, form a sum graph. Sum graph labellings of wheels have been examined previously by other investigators. The sum number of an n spoked wheel, σ(Wn), n⩾5, has been found to be n when n is odd and n2+2 when n is even; however the proofs of these results are incomplete. In this paper we provide a complete proof and affirm that these previous results are correct.