Some New Results On Mod Sum Graph And Integral Sum Graph

Graph theory is a branch of mathematics, especially an important branch of discrete mathematics. It has been applied in many different fields in the modern world, such as physiccs, chemistry, astronomy, geography, biology, as well as in computer science and engineering.Most graph labelling methods origin to one paper introduced by Rosa in 1967.i.e. 《 On certain valuation of the vertices of a graph 》 . A vertex labelling of a graph G is an assignment f of label to the vertices of G that induces for each edge uv a label depending on the vertex labels f(u) and f(v) In 1988, Harary introduced the notion of a sum graph. A graph G(V, E) is called a sum graph if there is an injective labelling / from V to a set of positive integers S such that uv 6 E if and only if f(u) + f(v) ∈ S. Since the vertex with the highest label in a sum graph cannot be adjacent to any other vertex, every sum graph must contain isolated vertices. For a connected graph G, let cr(G) denote the minimum number of isolated vertices that must be added to G so that the resulting graph is a sum graph.In 1990, J. Boland, R. Laskar, C. Turner and G. Domke investigated a modular version of sum graphs. They call a graph G{V, E) a mod sum graph if there exists a positive integer n and an injective labelling from V to {1,2, ? ? ? , n - 1} such that uv ∈ E if and only if f(u) + f(v)(mod n) = f(w) for some vertex w, obviously, all sum graphs are mod sum graphs. However, not all mod sum graphs are sum graphs. For a connected graph G which is not a mod sum graph, let p(G) denote the minimum number of isolated vertices that must be added to G so that the resulting graph is a mod sum graph. It is called the mod sum number of G.In 1994, Harary generalized sum graphs by permitting S to any set of integers. He called these graphs integeral sum graphs. Obviously, not all connected graphs are integral sum graphs. For a connected grpah G which is not an integral sum graph, let ξ(G) denote the minimum number of isolated vertices that must be added to G so that resulting graph is an integral sum graph. It is called the integral sum number of G.The main results obtained in this thesis can be summarized as follows:1. Prove relevant concepts and the basic theorems on sum graph;2. In this article chapter three section two, we will show a new type of mod sum graphs < C₄;n >, < C₅;n > and < C₆;n >;3. In this article chapter four section two, we use the idea of identification to prove that lobster trees and flower trees are integral sum graphs;4. In this article chapter four section three, we will show that MS{mn} are integral sum graphs.