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 ∈ E if and only if u + v∈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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics