Some New Properties of Cyclotomic Cliques Arrangements

In this paper, we study the arrangement of lines in the euclidean plane constructed from the geometric clique graph generated by the regular -gon. The vertices of this clique arrangement are located on finite concentric circles that we call orbits. We focus especially on the number of orbits and the number of vertices inside and outside the regular -gon. Combinatorics in finite sets of quadruplets of integers provide information on the way the orbits are distributed. Next, using cyclotomic fields, we give galoisian properties of the radii of the orbits and their cardinalities. Keywords: arrangement of lines in the plane, cyclotomic fields, geometric graphs, Galois theory