A natural graph of finite fields distinguishing between models

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. (c) 2020 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This study establishes a graph structure associated with finite fields, distinguishing between models of isomorphic fields and interpreting basic notions in terms of standard graph properties. The connectedness and diameter estimation of these graphs are discussed, along with an accidental graph isomorphism. The focus is also on the non-trivial Laplace eigenvalue in a specific family of 8-regular graphs, demonstrating it is not an expander. Additionally, a regular covering graph is introduced, showing connectivity only when the root is primitive.