The Study On The Integral Sum Number Of Trees

Frank Harary introduced the concepts of sum graphs and integral sum graphs[ 13] [15]. Plenty of research and exploration has been done on this field in the last ten years. Many achievements have been gained~ meanwhile, a considerable number of unsolved problems have been remained. In this paper first we systematically summarize the results of the research on sum graphs and integral sum graphs in the last ten years. Then we fully detail the Ellingham抯 labelling algorithm[8), which can be used to proved that any tree other than K1 can be made into a sum graph with the addition of a single isolated vertex. On the base of it, we successfully extend the sum labelling of trees into the integral sum labelling of trees by introducing the new concepts of hanging path and tail. Thus we obtain the result on the integral sum graphs, that is, any tree with tail at least length 3 is an integral sum graph. The result improves the preceding result of the integral sum trees from identification [17]. In addition, we prove that double stars are integral sum graphs~ thus overthrow a result in [15), both S(1, 3) and S(2, 2) are not integral sum graphs. We also point out that the integral sum labelling of the double star is unique in the isomorphic sense.