We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest non-trivial Laplace eigenvalue is given some attention, in particular for a specific family of 8-regular graphs showing that it is not an expander. We introduce a regular covering graph and show that it is connected if and only if the root is primitive.

中文翻译：

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区分模型的有限域自然图

我们定义了一种以自然方式与有限域相关联的图结构，该域仍然区分同构域的不同模型。有限域理论中的某些基本概念具有关于标准图属性的解释。我们证明了这些图是连通的，并提供了它们的直径的估计。一个偶然的图同构被发现并证明。最小的非平凡拉普拉斯特征值受到关注，特别是对于特定的8正则图族，表明它不是扩展器。我们引入一个规则的覆盖图，并表明只有在根是原始的情况下它才是连接的。