【作者】 魏建新；

【导师】 高敬振；

【作者基本信息】 山东师范大学 ， 应用数学， 2005， 硕士

【摘要】 本文仅考虑有限无向简单图,所用图论记号及术语遵循文献[1]. 1990年,F.Harary提出了和图的概念,令N表示正整数集,N的非空有限子集S的和图G+（S）是指图（S,E）,其中uv∈E当且仅当u+v∈S,一个图G称为和图,若它同构于某个S（?）N的和图,此时我们说S给出了G的一个和标号。而一个图G的和数σ（G）是使得G∪nK₁是和图的非负整数n的最小值。 1994年,F.Harary又介绍了整和图、整和数的概念,即把和图、和数定义中的正整数集N换成整数集Z。 模和图的概念是由Boland等人提出的。模和图是取S（?）Z_m\{0}且所有算术运算均取模m（≥|S|+1）的和图,其中Z_m={0,1,2,…,m-1},一个图G的模和数ρ（G）是使得G∪ρK₁是模和图的孤立点数ρ的最小值,这个概念是Sutton等人提出来的。 从实用的角度来看,（整,模）和图标号可用作图的压缩表示,即表示图的数据结构,可作为图的一种定义及存储方式。 目前对和图的研究主要集中在两个方面:一方面研究图的（整）和数与其它图参数、图结构的联系;另一方面是从一些特殊图类着手,确定它们的和数、整和数与模和数。迄今为止,已经取得了许多成果,提出了一些好的方法,给出了一些一般性的理论,并且还将和图的概念推广到了超图上。 在本文的第一章中,我们主要介绍了关于和图的一些概念、术语、符号,而且给出了几个关于（整）和图与其它图参数、图结构联系方面的定理;在第二章确定更多还原

【Abstract】 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更多还原