The Mod Sum Graph And Directed Sum Graph

Harary presented the concept of sum graph in 1990. Harary 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 say that S is an (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 result in an (integral) sum graph.The concept of the mod sum graph was introduced by Boland etal in 1990. A mod sum graph is a sum graph with S (?) Z_m\{0} and all arithmetic performed modulo m where m≥|S|+1. Then. Sutton etal presented the concept of mod sum number. The mod sum numberρ(G) of G is the least numberρof isolated verticesρK₁ such that G(?)ρK₁ is a mod sum graph.Miller etal [6] presented the concept of exclusive graph in 2003. A sum labeling S is called an exclusive sum labeling. ifμ+v∈S\V(G) for any edge uv∈E(G). The, exclusive sum numberε(G) of G is the least numberεof isolated verticesεK₁ such that G(?)εK₁ is an exclnsive, sum graph. Dou and Gao presented the concept of mod integral sum graph in 2007. A mod integral sum graph is a sum graph with S(?) Z_m and all arithmetic performed modulo m where m≥|S|. The mod integral sum numberΨ(G) of G is the least numberΨof isolated verticesΨK₁ such that G (?)ΨK₁ is a mod integral 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.New concepts are presented in this thesis as follows:The directed sum graph (?)+(S) of a nonempty finite subset S(?)N is the digraph (S. E) with (?)∈E if and only if u-v∈S. A digraph G is said to be a directed sum graph if it is isomorphic to the directed sum graph of some S(?)N. The directed sum number (?) (G) is the smallest number of isolated vertices which when added to G result in a directed sum graph.A directed sum labeling S is called an exclusive directed sum labeling. if u-v∈S\V(G) for any edge (?)∈E(G).Direct sum labeling and exclusive directed sum labeling are another ways used as compressed representation of a digraph.The first chapter of this paper gives a brief introduction about the basic concepts, terminologies and symboles which are used in this paper: In the second chapter we give some structual results on mod sum graphs: In the third chapter. we determine a type of graphs which are not mod sum graph: In the fouth chapter. we introduce the concept of directed sum graph and exclusive directed sum graph and determine the (directed, exclusive directed) sum number of some digraph: At last, in the fifth chapter we advance the things to be discussed.In this paper we have the following theorems. Theorem 2.1.1 The mod sum graph has at most one saturated point.Corllary 2.1.1 The saturated vertex of mod sum graph G with order n can be expressed by km/(n-1), where 1≤k≤n-2 and m is the modulo.Theorem 2.2.1 If the saturated vertex of graph G with saturated degree is prime and more than 2, then G is not a mod sum graph except the star and K_1,3+ e.Theorem 2.3.1 The union of a sum graph and a mod sum graph is a mod sum graph.Theorem 2.3.2 The union of the mod sum graph G, H with mod labelling S₁, S₂ and the modulus z₁, z₂, (z₁, z₂)=1, then G(?)H is a mod sum graph.Theorem 2.4.1 Unit sum graph G is a mod sum graph, if its second label is not adjacent to its minium label, then G.Theorem 2.4.2 There must be a vertex in the unit sum graph with which when identified a star results in a mod sum graph.Theorem 2.4.3 If the sum of any two nonadjacent labels of the mod graph G is not the modulus, then K₁∨G is a mod integral sum graph.Theorem 2.4.4 There must be a r-regular mod sum graph for any positive integer r.Theorem 3.1.1 F_n.m is not a mod sum graph.Theorem 3.2.1 W_n.m is not a mod sum graph.Theorem 4.1.1 If G is a directed sum graph, them G(?)mk₁ is a directed sum graph too, for any m.Theorem 4.1.2 If G(?)mK₁ has an exclusive directed sum labelling, then G(?)(m + q)K₁ has an an exclusive directed sum labelling too, for any q.Theorem 4.2.3 If both (?)(G₁) and (?)(G₂) exist, then (?) (G₁(?)G₂)≤(?)(G₁)+ σ(G₂)Theorem 4.3.1 m(?) is a directed sum graph.Theorem 4.3.2 The directed path (?) is a directed sum graph.Theorem 4.3.3 (?)=1.Theorem 4.3.4 (?)=1.