Graphical Arrangement

Let be a simple graph on the vertex set . The graphical arrangement of is the collection of hyperplanes in given by

(1)

Thus every subarrangement of the braid arrangement is graphical. The complete graph gives the full braid arrangement, while the empty graph gives the empty arrangement.

The arrangement characteristic polynomial equals the chromatic polynomial of the graph,

(2)

For example, a tree on vertices has .

Nian et al. (2026) associate a finite-field arrangement with . Let be a prime power, let be the finite field of order , and let be the collection of nonempty cliques of . Then

(3)

Coefficient vectors differing by a nonzero scalar define the same hyperplane. For every nonnegative integer , its characteristic polynomial satisfies

(4)

Consequently, if , the characteristic polynomial of determines the chromatic polynomial of .

An arrangement is called free when its module of logarithmic polynomial vector fields is a free module. The graph is chordal graph iff both and are free arrangements. Suppose is a perfect elimination ordering, meaning that the earlier neighbors of each form a clique, and let be the number of those neighbors. Then

(5)

In particular, for a path graph , this is .