Some Results On (Mod, Integral) Sum Graph

All graphs considered in this thesis are finite,simple and undirected. We follow in general the graph-theoretic notation and terminology in[1].The notion of sum graph was introduced by F.Harary [2] 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 (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 isomorphic to the sum graph of some S(?) N.In this case we say that 5 gives an sum labelling for G. The sum number σ(G) of G is the smallest number of isolated vertices which when added to G result in a sum graph.In 1994, F.Harary [3] introduced the concepts of integral sum graph and integral sum number of a graph with S C Z(the set of all integers) instead of S(?) N.Mod sum graph was introduced by Boland et al.[4]. A mod sum graph is a sum graph with S (?) Zm\{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 pK1 such that G∪ pK1 is a mod sum graph. This concept was introduced by Sutton et al.[5]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, and an alternative method for defining and storing graphs.Now the research aims at two aspects. One is to study the relation between the (integral) sum number and other parameters and structures of the graph. The other is at determining the sum number, integral sum number and mod sum number of some graph classes. Some achievements have been gotten , some good method and general theorems have been gotten ,and have been extended to hypergraphs.The first chapter of this thesis gives a brief introduction about the basic concepts, terminologies and symboles which are used in this thesis, and surveys some related results about (integral) sum graphs. In the second chapter we determine the sum number of 2-regular graph,crown Cn ⊙ K1, incomplete crown graph C'n ⊙ K1, the subdivision graph of Cn⊙K1 and complete crown graph Cn⊙K1* and mod sum...