Generating functions of $q$-chromatic polynomials

Abstract

Given a graph $G=(V,E)$ and a linear form $\lambda \in \mathbb{Z}_{ > 0 }^V$, Bajo et al. (2025) introduced the $q$-chromatic polynomial $\chi_G^\lambda(q,n) := \sum q^{\sum_{v \in V} \lambda_v c(v)}$ where the sum is over all proper colorings $c: V \to [n] := \{ 1, 2, \dots, n \}$; they showed that $\chi_G^\lambda(q,n)$ is a polynomial in $[n]_q := 1 + q + \dots + q^{ n-1 } $ with coefficients in $\mathbb{Z}(q)$. For $d \in \mathbb{Z}_{>0}$ and the linear form given by $(d,d^2,\ldots,d^d)$, we show that the $q$-chromatic polynomial distinguishes labeled graphs with vertex set $[d]$. Using permutation statistics introduced by Chung--Graham (1995), called $G$-statistics, and polyhedral geometry, we give the multivariate integer point transform for the region of proper colorings of a given graph $G$. This integer point transform allows us to find the generating function for the $q$-chromatic polynomial with respect to any linear form. We further specialize these results to the linear form $(1, 1, \dots, 1)$, which allows us to write the $q$-chromatic polynomial in the $q$-binomial basis, clarifying expressions found by Bajo et al. Moreover, we show that $G$-statistics are compatible with the theory of order polytopes used by Bajo et al. and Chow (1999). This yields further properties for the generating function of $q$-chromatic polynomial with linear form $(1, 1, \dots, 1)$, where certain coefficients of the numerator polynomial are palindromic polynomials in $q$.