Galois Graphs: Walks, Trees and Automorphisms

TLDR: In this article, a Galois graph G(Φ(x, y) is defined by taking the algebraic closure as the vertex set and adjacencies corresponding to the zeroes of the symmetric polynomial over a perfect field k of characteristic zero, and some graph properties such as lengths of walks, distances and cycles are described in terms of Φ.

Abstract: Given a symmetric polynomial Φ(x, y) over a perfect field k of characteristic zero, the Galois graph G(Φ) is defined by taking the algebraic closure as the vertex set and adjacencies corresponding to the zeroes of Φ(x, y). Some graph properties of G(Φ), such as lengths of walks, distances and cycles are described in terms of Φ. Symmetry is also considered, relating the Galois group Gal( ) to the automorphism group of certain classes of Galois graphs. Finally, an application concerning modular curves classifying pairs of isogeny elliptic curves is revisited.