The (integral) sum number of Kn−E(Kr)

The concept of the (integral) sum graphs was introduced by Harary (Congr. Numer. 72 (1990) 101; Discrete Math. 124 (1994) 99). Let N(Z) denote the set of all positive integers(integers). The (integral) sum graph of a finite subset S⊂N(Z) is the graph (S,E) with two vertices that are adjacent whenever their sum is in S. A graph G is said to be a (integral) sum graph if it is isomorphic to the (integral) sum graph of some S⊂Z. The (integral) sum number of a given graph G, denoted by σ(G)(ζ(G)), was defined as the smallest number of isolated vertices which when added to G resulted in a (integral) sum graph. In this paper, the integral sum number and the graph (Kn−E(Kr)), which is an unsolved problem proposed by Harary (1994) in 1994, are investigated and determined. The results on the integral sum number of graph (Kn−E(Kr)) are presented as follows:ζ(Kn−E(Kr))=0(r=n,n−1),n−2(r=n−2),n−1(n−3⩾r⩾⌈2n3⌉−1),3n−2r−4(⌈2n3⌉−1>r⩾n2),2n−4(⌈2n3⌉−1⩾n2>r⩾2),where n⩾5,r⩾2. Furthermore, if r≠n−1, then σ(Kn−E(Kr))=ζ(Kn−E(Kr)).