All Cayley representations of the distant graph \(\Gamma _{\mathbb Z}\) over integers are characterized as Neumann subgroups of the extended modular group. Possible structures of Neumann subgroups are revealed and it is shown that every such structure can be realized.

中文翻译：

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Neumann子群的Cayley图

整数上的远图\（\ Gamma _ {\\ mathbb Z} \）的所有Cayley表示均表示为扩展模块化组的Neumann子组。揭示了诺伊曼子群的可能结构，并表明可以实现每种这样的结构。