Hararyʹs conjectures on integral sum graphs

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S ⊂ N(Z) is the graph (S, E) with uv ϵ E if and only if u + v ϵ S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S ⊂ N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n ⩾ 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.