Some Results On Sum Graph Theory And Fractional Graph Theory

In this thesis we study graphs from two aspects and obtain some results. The first part of this thesis is about sum graphs. Hararv[1] presented the concept of sun graph in 1990.Let V(G) denote the vertex set of a graph G and |S| denote the number of elements in 5. Let N(Z) denote the set of all positive integers (integers). The sum graph G+(S) of a nonempty finite subset S C 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 sum graph of some S (?) N(Z). The (integral) sum number σ(G)(ζ(G)) is the smallest number of isolated vertices which when added to G result, in an (integral) sum graph. A mod sum graph is a sum graph with .S (?) Zm\{0} and all arithmetic performed modulo in where in > |S|+1. The mod sum number p(G) of G is the least, number p of isolated vertices pK1 such that G∪pK1 is a mod sum graph. 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 number ε(G) of G is the least number ε of isolated vertices εK1 such that G ∪ εK1 is an exclusive sum graph.From a practical point of view, sum graph labelling can be used as a compressed representation of a graph, a data structure for representing the graph. Data compression is important not only for saving memory space but also for speeding up some graph algorithms when adapted to work with the compressed representation of the input graph.Now the research work for sum graph thory aims at, determining the sum number, integral sum number and mod sum number of some graph classes. In the second chapter of this thesis we discuss the exclusive sum number for disjoint union of two graphs and the exclusive sum number of some graph classes; In the third chapter we discuss the sum number, integral sum number and mod sum number of graph Kr,s - E(rK2) and determine the sumnumber of subdivided-wheel Wn,1 and subdivided-fan Fn,1 , and we have gotten some achievements.Theorem 2.1.1 Let G1,G2 be two graphs, Li be an exclusive sum labeling of Gi∪ ε(Gi)K1 and Ci be the isolated set of Li(i = 1,2). If maxC1 and minC2 are mutually prime, then ε(G1 ∪ G2) ≤ ε(G1)+ ε(G2) - 1.Corollary 2.1.1 Let Gi be an exclusive graph and Ci be the isolated set of Gi∪ε(Gi)K1(i = 1,2). If max C1 and min C2 are mutually prime, then σ(G1∪G2) ≤ σ(G1) + σ(G2)-1.Theorem 2.2.1 For s≥ r, ε(Kr,s) = s + r -1.Theorem 2.2.2 For s≥ r,Theorem 2.2.3 For n ≥ 5,Theorem 2.2.4 3 ≤ ε(Cn ⊙ K1) ≤ 4 (n ≥ 3).Theorem 3.1.1 σ(Wn,1) = 2 for n ≥ 3.Theorem 3.1.2 a(F;iil) = 2 for n > 3.Theorem 3.1.3 For the graph K3,4 - Es(3K2) ( i.e. W3,1). σ = p = ζ= 2.Theorem 3.2.1 For s > rTheorem 3.3.1 sl + r-3 < ζ(Kr,s-E(rK2)) ≤ σ{Kr,s-E(rK<2)) < tk + r-3 for .s > r ≥ 4 and (r, s) ≠ (4,5), whereCorollary 3.3.1 ζ(Kr,s - E(rK2)) = σ(Kr,s - E(rK2)) = s+3r-7/2 for 4 (dc{x) -1)g(y) for any x,y ∈ V and x ≠y, then the graph G - e is fractional (g,f)-deleted for any edge e ∈ E.