In Graph Theory, Labeling Problem

Most graph labeling methods trace origin to one introduced by Rosa in 1967.i.e (KOn certain valuations of the vertices of a graph~) .A vertices of a graph G is an assingnment f of label to the vertices of G that induces for each edge xy a label depending on the vertex labels f(x) and f(y).The two best konwn labeling methods are calls graceful and harmonious labeling.A function f is calls a graceful labeling of a graph G with q edges if f is an injection from the vertices of G to set {0,l,~~,q} such that ,when each edge xy is assigned the label If(x-f(y)I,the resulting edge labels are distinct. A function f is calls harmonious if it is an injection from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y) (mod q),the resulting edge labels are distinct. In 1990, Harary introduced the notion of a sum graph .A graph G(V,E) is called a sum graph if there is an bijective labeling f from V to a set of positive integers S such that xy 6 E if and only if f(x)+f(y) ES .Jn 1994 Harary generalized sum graphs by permitting S to any set of integers .,He calls these graphs integral sum graphs. In 1970 Kotzig and Rosa defined a magic labeling of a graph G(V,E) as a bijection f from VUE to {1,2,~,IVUEI}such that for all edges xy, f(x)眆(y)+f(xy) is constant.In this paper ,applying the methods of identification, we prove the conjecture which every tree is an integral sum graph by chen and every tree is edge-magic.Lastly we give a kind of non-harmonions.