The (Mod, Integral, Mod Integral)sum Numbers Of Several Kinds Of Graphs

The notion of sum graph was introduced by F.Harary in 1990. Let N denote the set ofall positive integers. The sum graph G⁺ (S) of a finite subset S ? N is the graph (S,E)with uv∈E if and only if u +v∈S . A graph G is said to be a sum graph if it is isomorphicto the sum graph of someS ? N . In this case we say that S gives an sum ymbols for G.The sum number σ(G) of G is the smallest number of isolated vertices which when added toG result in a sum graph.In 1994, F.Harary introduced the concepts of integral sum graph and integral sumnumber of a graph with S ? N (the set of all integers) instead of S ? N .Mod sum graph was introduced by Boland et al. A mod sum graph is a sum graphwith S ? Z_m \{0} and all arithmetic performed modulo m wherem ≥ |S| +1. The modsum number ρ(G) of G is the least number ρ of isolated vertices ρK₁ such thatG ∪ ρK₁ is a mod sum graph. This concept was introduced by Sutton et al.[4]From a practical point of view, All kinds of sum graph ymbols can be used as acompressed representation of a graph, a data structure for representing the graph. Datacompression is important not only for saving memory space but also for speeding up somegraph algorithms when adapted to work with the compressed representation of the inputgraph.The first chapter of this paper gives a brief introduction about the basic concepts,terminologies and ymbols which are used in this paper.BajiaofanT_nis a graph of adding ahung side in the axis K₁ of fanF_n = P_n ∨ K₁.UmbrellaJ_nis a graph of adding a hung sidein the axis K₁ of wheelW_n = C_n ∨ K₁. In second and third chapters we discuss respectively thesum number, integral sum number, mod sum number and mod integral sum number of T_n ,J_n and the subdivision graph of T_n and J_n .In forth chapter, we determine the integral sumnumber of the subdivision graph Gn *of jointed circles Cn × K2 ,and the sum number of thelantern and incomplete lantern ,and also proves the windmill is the integral sum graph,and forthe ladder Ln = Pn × K2, KL3 is the mod integral graph.In this thesis we obtain the following theorems:Theorem 2.1.1 For n ≥ 3, ρ (T n) = 1.Theorem 2.2.1 Tn ( n ≥ 3)is the integral sum graph and mod integral sum graph.Theorem 2.2.22, 4( ) 3, 3 64, 5 .nnT n or n evenn oddσ≤ ???= =≥?? ≥,, .Theorem 2.3.1 T2 * is the mod sum graph,andσ (T n * ) ??? ≤= 12,, nn=≥24, 3.,Theorem 3.1.1 For n ≥6 even, ρ ( Jn)=1.Theorem 3.2.1 Umbrella J n( n ≠3) is the integral sum graph,and also is the mod integralsum graph.Theorem 3.2.2 For n ≥ 2, σ ( Jn*) ≤ 2.Theorem 4.1.1 For n ≥ 2, ζ (G n*) ≤ 5.Theorem 4.2.1 kL3 ( k ≥ 2) is the mod sum graph, and also is the mod integral sum graph.Theorem 4.2.2 For n ≥ 3, σ ( L*n ) = 2 or3.Theorem 4.3.1 Windmill W n*( n ≥2) is the integral sum graph, and also is the modintegral sum graph.Theorem 4.4.1 Lantern Bn( n ≥2) is the integral sum graph, mod sum graph, and also isthe mod integral sum graph.Theorem 4.4.2 The sum number of incomplete lantern B n*( n ≥3) is 1.