The (Exclusive, Exclusive Integral, Lower Integral) Sum Numbers Of Several Kinds Of Graph

All graph considered in this paper are finite,simple and undirected.We follow in general the graph-theoretic notation and terminology of [1].Harary[2] presented the concept of sum graph in 1990. Harary[3] presented the concept of integral sum graph in 1994. Let N( Z ) denote the set of all positive integers (integers). The (integral ) sum graph G+(S) of a nonempty 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 ) . We said that S is one of the (integral) sum labeling,and we consider the vertices and labelings as the same.The (integral) sum numberσ(G)(ζ(G)) is the smallest number of isolated vertices which when added to G resulted in an (integral) sum graph.Miller[4] presented the concept of Exclusive graph in 2003. A sum labeling S is called an exclusive sum labeling , if u+v∈S\ V(G) for any edge uv∈E(G). The exclusive sum (integral sum) numberε(G) (ζ′( G))of G is the least number which when added to G resulted in is an exclusive sum (integral sum) graph.In 2004,Li min [5] introduced the concept of lower integral sum graph. Let Q+ denote the set of all the positive rational number. The lower integral sum graph G+ (S) of a nonempty finite subset S? Q+ is the graph (S,E) with uv∈E if and only if ?u + v?∈S. A graph G is said to be an lower integral sum graph if it is isomorphic to the lower integral sum graph of some S? Q+. The lower integral...