Some Results on Sum Graph, Integral Sum Graph and Mod Sum Graph 
Received:March 18, 2005 Revised:April 25, 2005 
Key Words:
sum graph integral sum graph mod sum graph flower tree Dutch $m$windmill.

Fund Project: 
Author Name  Affiliation  ZHANG Ming  Department of Applied Mathematics, Dalian University of Technology, Liaoning 116024, China School of Energy and Power Engineering, Dalian University of Technology, Liaoning 116024, China  YU Hongquan  Department of Applied Mathematics, Dalian University of Technology, Liaoning 116024, China  MU Hailin  School of Energy and Power Engineering, Dalian University of Technology, Liaoning 116024, China 

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Abstract: 
Let $N$ denote the set of positive integers. The sum graph $G^{+}(S)$ of a finite subset $S\subset N$ is the graph $(S,E)$ with $uv\in E$ if and only if $u+v\in S$. A graph $G$ is said to be a sum graph if it is isomorphic to the sum graph of some $S\subset N$. By using the set $Z$ of all integers instead of $N$, we obtain the definition of the integral sum graph. A graph $G=(V,E)$ is a mod sum graph if there exists a positive integer $z$ and a labelling, $\lambda$, of the vertices of $G$ with distinct elements from $\{0,1,2,\ldots,z1\}$ so that $uv\in E$ if and only if the sum, modulo $z$, of the labels assigned to $u$ and $v$ is the label of a vertex of $G$. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch $m$windmill ($D_{m}$) is integral sum graph and mod sum graph, and give the sum number of $D_{m}$. 
Citation: 
DOI:10.3770/j.issn:1000341X.2008.01.028 
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