Galois Graphs: Walks, Trees and Automorphisms

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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 \(\bar k\) 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(\(\bar k\)/k) to the automorphism group of certain classes of Galois graphs. Finally, an application concerning modular curves classifying pairs of isogeny elliptic curves is revisited.