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On Single-Distance Graphs on the Rational Points in Euclidean Spaces
Canadian Mathematical Bulletin ( IF 0.6 ) Pub Date : 2020-07-29 , DOI: 10.4153/s0008439520000181
Sheng Bau , Peter Johnson , Matt Noble

For positive integers n and d > 0, let $G(\mathbb {Q}^n,\; d)$ denote the graph whose vertices are the set of rational points $\mathbb {Q}^n$, with $u,v \in \mathbb {Q}^n$ being adjacent if and only if the Euclidean distance between u and v is equal to d. Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $\mathbb {Q}^n$. In this paper, we show that a space $\mathbb {Q}^n$ has the property that all pairs of non-trivial distance graphs $G(\mathbb {Q}^n,\; d_1)$ and $G(\mathbb {Q}^n,\; d_2)$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $G(\mathbb {Q}^n,\; d)$.



中文翻译:

欧氏空间中有理点的单距离图

对于正整数nd > 0,让$ G(\ mathbb {Q} ^ n,\; d)$表示图的顶点是有理点集合$ \ mathbb {Q} ^ n $,其中$ u ,v \ in \ mathbb {Q} ^ n $当且仅当uv之间的欧几里得距离等于d时才相邻。如果d实际上被实现为$ \ mathbb {Q} ^ n $点之间的距离,则该图被认为是“非平凡的” 。在本文中,我们证明了空间$ \ mathbb {Q} ^ n $具有以下性质:所有成对的非平凡距离图$ G(\ mathbb {Q} ^ n,\; d_1)$当且仅当n等于1、2或4的倍数时, $ G(\ mathbb {Q} ^ n,\; d_2)$是同构的。的$ G(\ mathbb {Q} ^ n,\; d)$

更新日期:2020-07-29
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