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 $m$-wind-mill.
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 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 Hong-quan Department of Applied Mathematics, Dalian University of Technology, Liaoning 116024, China MU Hai-lin School of Energy and Power Engineering, Dalian University of Technology, Liaoning 116024, China
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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 $\{0,1,2,\ldots,z-1\}$ 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 $m$-wind-mill ($D_{m}$) is integral sum graph and mod sum graph, and give the sum number of $D_{m}$.