$q$-deformation of chromatic polynomials and graphical arrangements
Tongyu Nian, Shuhei Tsujie, Ryo Uchiumi, Masahiko Yoshinaga
TL;DR
This work develops a $q$-analogue of graphical arrangements by introducing $\\mathcal{A}_G^q$, a subarrangement of the finite-field hyperplane arrangement that mirrors the Vandermonde–Moore determinant correspondence. It proves a $q$-deletion–contraction framework, shows that for large $q$ the $q$-deformed characteristic polynomial determines the graph’s chromatic polynomial and connects to stable partitions, and establishes freeness equivalence with chordality, providing an explicit basis for the logarithmic derivations in the chordal case. The results fuse graph-coloring invariants with hyperplane-arrangement theory through $q$-analogues, offering concrete formulas for paths and cycles and a structural criterion for freeness via perfect elimination orderings. Overall, the paper extends combinatorial invariants into a finite-field geometric setting, enriching both arrangement theory and graph coloring with new algebraic tools and identities.
Abstract
We first observe a mysterious similarity between the braid arrangement and the arrangement of all hyperplanes in a vector space over the finite field $\mathbb{F}_q$. These two arrangements are defined by the determinants of the Vandermonde and the Moore matrix, respectively. These two matrices are transformed to each other by replacing a natural number $n$ with $q^n$ ($q$-deformation). In this paper, we introduce the notion of ``$q$-deformation of graphical arrangements'' as certain subarrangements of the arrangement of all hyperplanes over $\mathbb{F}_q$. This new class of arrangements extends the relationship between the Vandermonde and Moore matrices to graphical arrangements. We show that many invariants of the ``$q$-deformation'' behave as ``$q$-deformation'' of invariants of the graphical arrangements. Such invariants include the characteristic (chromatic) polynomial, the Stirling number of the second kind, freeness, exponents, basis of logarithmic vector fields, etc.