Two New Classes Of Integral Sum Graphs

The concept of the (integral)sum graphs was introduced by Harary. let N* denote the set of positive integers, The(integral)sum graph G+(S) 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 called an(integral)sum graph if it is isomorphic to the (integral)sum graph G+(S) of some S (?) N*(Z). The (integral)sum number of a given graph G is the smallest number of isolated vertices which when added to G result in an (integral)sum graph. For convenience, an integral sum graph is written as ∫ ∑ -graph.This paper is made up of five parts. Firstly, Section One is an introduction to the backgroud of the research and the results we gain in this paper. Section Two and four are about some new classes of ∫ ∑ —graph. In section Three we prove that all the odd cycle are integral sum graph by the othe way. In the same time,we solve a problem posed by Baogen Xu in [3]. In section five we disprove two conjectures of Harary's.In addition, some figtures are given to explain the labelling intuitivly.