| 张明,于洪全,穆海林.和图、整和图与模和图的几个结果[J].数学研究及应用,2008,28(1):217~222 |
| 和图、整和图与模和图的几个结果 |
| Some Results on Sum Graph, Integral Sum Graph and Mod Sum Graph |
| 投稿时间：2005-03-18 修订日期：2005-04-25 |
| DOI：10.3770/j.issn:1000-341X.2008.01.028 |
| 中文关键词: 和图 整和图 模和图 花树 荷兰风车. |
| 英文关键词:sum graph integral sum graph mod sum graph flower tree Dutch $m$-wind-mill. |
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| 中文摘要: |
| 令$N$代表正整数,有限子集$S\subset N$的和图$G^{+}(S)$是$uv\in E$当且仅当$u+v\in S$.若图$G$同构与某一个$S\subset N$,则称$G$是和图.若用整数集$Z$代替$N$,则得到整和图.若存在一个正整数$z$,用$\{0,1,2,\ldots,z-1\}$代替$N$,运算加法取模$z$,则得到模和图.本文证明了花树是整和图、荷兰风车是整和图和模和图,并给出荷兰风车的和数. |
| 英文摘要: |
| 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 $D_{m}$. |
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