Generating functions of q-chromatic polynomials

Given a graph $G=(V,E)$ and a linear form λ∈ Z> 0V, Bajo et al. (2025) introduced the q-chromatic polynomial χG^λ(q,n) := ∑ q^∑v ∈ V λᵥ c(v) where the sum is over all proper colorings c: V → [n] := \ 1, 2, , n \; they showed that χG^λ(q,n) is a polynomial in [n]q := 1 + q + + qⁿ⁻¹ with coefficients in Z(q). For d ∈ Z>0 and the linear form given by (d,d²,…,dᵈ), 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, , 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, , 1), where certain coefficients of the numerator polynomial are palindromic polynomials in q.