Graphs & digraphs (2nd ed.)
The characteristic ideal of a finite, connected, regular graph
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
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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 as the vertex setand adjacencies corresponding to the zeroes of Φ(x, y). Some graph properties of G(Φ), such as lengths of walks, distances andcycles 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 concerningmodular curves classifying pairs of isogeny elliptic curves is revisited.