The (Mod, Integral, Mod Integral) Sum Number Of Some Graphs

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.We have researched about sum graph since Harary presented the concept of sum graph in 1990. Now our research work aims at determining the sum number, integral sum number and mod sum number of some graph classes. And we have gotten some achievements.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 and third chapters we determine respectively the sum number, integral sum number and mod sum number of fan . In forth chapter we discuss the mod sum number of complete bipartite graphs and the graph . In the last chapter we define the concept of mod integral sum graph and mod integral sum number, give some relations of mod sum number, integral sum number and mod integral sum number, and discuss the mod integral sum number of some graph classes.Let V(G] denote the vertice set of the graph G and \S\ denote the number of the elements in 5. Let N(Z] denote the set of all positive integers (integers). The sum graph G+(S) of a finite subset S C N(Z] is the graph (S,E) with uv 6 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 5 C N(Z). The (integral) sum number er 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 5 C Zm\{0} and all arithmetic performed modulo m where m > (51 + 1. The mod sum number p(G) of G is the least number p of isolated vertices pK\ such that G U pK\ is a mod sum graph.In this paper we have the following theorems.Similarly we can define the concepts of mod integral sum graph and mod integral sum number. A mod integral sum graph is a sum graph with Sc Zm and all arithmetic performed modulo m where m > \S\. The mod integral sum numberof graph G is the least number ip of isolated vertices tyK\ such that is a mod integral sum graph.Lemma 5.1.1 For any graph G if there doesn't exist some vertex with degree...