  For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we
introduce the notion of Q-datum, which generalizes the notion of a Dynkin
quiver with a height function from the viewpoint of Weyl group combinatorics.
Using this notion, we develop a unified theory describing the twisted
Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh,
J. Algebra, 2019] with an appropriate notion of the generalized Coxeter
elements. As a consequence, we obtain a combinatorial formula expressing the
inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the
result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced
case. We also find several applications of our combinatorial theory of Q-data
to the finite-dimensional representation theory of the untwisted quantum affine
algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of
the quantum Cartan matrix, (i) we give an alternative description of the block
decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and
[Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially
conjectural) formula of the denominators of the normalized R-matrices between
all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants
$\Lambda(V,W)$ and $\Lambda^\infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park,
Compos. Math., 2020] for each pair of simple modules $V$ and $W$.
