(Lower-integral) Sum Graph Structure And (Lower-integral) Sum Number Of Some Graphs

All graphs considered in this paper are finite,simple and undirected. We follow in general the graph-theoretic notation and terminoligy of[1].The concept of sum graph,sum number was discovered by F.Harary in 1990. Let N denote the set of all positive integers . The sum graph G⁺(S) of a finite subset S C N is the graph (5, E) with uv G E if and only if u + v ∈ S. A graph G is said to be an sum graph if it is isomorphic to the sum graph of some S (?) N.And we say that S gives an sum labelling for G. The sum number σ(G) is the smallest number of isolated vertices which when added to G result in an sum graph.As we know that the minimum degree δof graph is the lower bound of its sum number,the one achieves this lower bound is called δ—optimal.In 1994, F.Harary also introduced the concept of the integral sum graph and integral sum number , those concepts are defined as the sum graph , the difference being that S (?) Z(the set ofall integers) instead of S (?) N.Mod sum graph was introduced by Boland et al.. A mod sum graph is a sum graph with S (?) Z_m\{0} and all arithmetic performed modulo m where m ≥ |S| + 1. The mod sum number p(G) of G is the least number p of isolated vertices pK\ such that G∪ ρK₁ is a mod sum graph. This concept was introduced by Sutton et al.In 2004, Li min ^introduced the concept of lower sum graph.Let Q+ denote the set of all positive rational number ,the lower-integral sum graph G+(s) of a finite subset S C Q+ is the graph (S, E) with uv ï¿¡ E if and only if \u + v\ 6 S.From a practical point of view,sum graph labelling can be used as a data structure for representing the graph,when input a graph with a sum graph labelling,it is also save saving spaces more than other labellings.Now our research work aims at two aspects. One is to study the relation between the sum number and other parameters and the structure of (integral) sum graphs. The other is at determining the sum number, integral sum number ,mod sum number and lower sum number of some graph classes. And we have gotten some achievements, alse gives some good method and generalized theorems and have extended the concepts to hypergraphs.The first chapter of this paper gives a brief introduction about the basic concepts, terminologies and symboles which are used in this paper;In second chapters we give some structual results on (lower integral)sum graph. In third chapter we discuss the lower integral sum numbers of paths,crowns and disjoint union of completed graphs.Fan Fn is the graph (V, E) with vertices set V = {c, a1;a-z,..-, an} and edge set E = {cai,ca2, ...,can,aia2, a2a3,..., an₁an},and every edge cai(i = l,,,n) is called spoke.We call spoke co, working spoke if caj,satisfied [c + 1, alm — c.Caterpilla is a tree that becomes a path when it is deleted its leaves (that is 1-degree vertex).if G\ and G% are graphs and G\ has n vertices the G\QG% is the graph obtained by taking one copy of G\ and n copies of G2 and joining the tth vertex of Gi withan edge to every vertex in the ith copy of G^.We call graph G\ ? G2 as crown.A tree T is said to be a three-path tree if it consists of three paths with a common end-note. It is a special case of a star-like tree.Such a tree is denoted by P(m, n, t) when its corresponding tree paths are Pm, Praand Pt, respectively.LetT = P(m,n,t), be any three-path tree,its corresponding three paths are written as follows: Pm - (ax, a2,..., am), Pn = (am, b2,..., bn),Pt = (am, c2,..., ct).In this paper we have the following theorems.Theorem 2.1.1 Let Li denotes the sum labellings of Gi [j a(Gi) (J K Gi,G2 are not both sum graphs, if there exists n 6 L2, and (max Li,n)=l, then a{G\JH) < a(G) + a{H) - I.Theorem 2.2.1 w^m / 4, 5) is 6-optimal.Theorem 2.3.1 In a lower integral sum labelling of Fn U a'(Fn)Ki(n > 5), there exists at least one spoke does not work.Theorem 2.4.1 If there exists a lower integral sum labelling S of three-path tree P(m,n,t)(m,n,t > 2),then the maximum integral labelling in S must be 1-degree vertex.Theorem 3.1.1 Any path is an lower integral sum graph.Theorem 3.2.1 Any caterpillar is an lower integral sum graph.Theorem 3.3.1 a(Cn 0 Kx) = l(n > 3).Theorem 3.4.1 a'(rKn) = l(n > 3,r > 2).Corolary 3.4.1 a'(Kni U Kn2 U ... U KUr) = l.here n{ > 3, r > 2,t = 1, ...r.