For d > 0, designate by G(Q3 , d) the graph whose set of vertices is the rational space Q3 , with any two vertices being adjacent if and only if they are Euclidean distance d apart. Deem such a graph to be “non-trivial” if d is actually realized as a distance between points of Q3 . In this work, we prove that non-trivial graphs G(Q3 , d1) and G(Q3 , d2) are isomorphic if and only if d1, d2 are rational multiples of each other. This determination of the isomorphism classes of graphs with vertex set Q3 answers a question posed by Johnson.
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机译：对于D> 0，通过G（Q3，d）表示顶点的图形是Rational Space Q3的图，其中任何两个顶点都是邻近的，如果它们是欧几里德距离D分开。如果D实际上被实现为Q3之间的距离，则认为这种图形是“非平凡”。在这项工作中，我们证明了IFONOMERPHIAL（Q3，D2）是彼此的合理倍数的非平移图G（Q3，D1）和G（Q3，D2）是同性的。这种确定与顶点集的图形的同构图Q3答案了约翰逊提出的问题。
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