  Phonon-mediated thermal conductivity, which is of great technological
relevance, fundamentally arises due to anharmonic scattering from interatomic
potentials. Despite its prevalence, accurate first-principles calculations of
thermal conductivity remain challenging, primarily due to the high
computational cost of anharmonic interatomic force constant (IFCs)
calculations. Meanwhile, the related anharmonic phenomenon of thermal expansion
is much more tractable, being computable from the Gr\"uneisen parameters
associated with phonon frequency shifts due to crystal deformations. In this
work, we propose a novel approach for computing the largest cubic IFCs from the
Gr\"uneisen parameter data. This allows an approximate determination of the
thermal conductivity via a much less expensive route. The key insight is that
although the Gr\"uneisen parameters cannot possibly contain all the information
on the cubic IFCs, being derivable from spatially uniform deformations, they
can still unambiguously and accurately determine the largest and most
physically relevant ones. By fitting the anisotropic Gr\"uneisen parameter data
along judiciously designed deformations, we can deduce (i.e., reverse engineer)
the dominant cubic IFCs and estimate three-phonon scattering amplitudes. We
illustrate our approach by explicitly computing the largest cubic IFCs and
thermal conductivity of graphene, especially for its out-of-plane (flexural)
modes that exhibit anomalously large anharmonic shifts and thermal conductivity
contributions. Our calculations on graphene not only exhibits reasonable
agreement with established DFT results, but also presents a pedagogical
opportunity for introducing an elegant analytic treatment of the Gr\"uneisen
parameters of generic two-band models. Our approach can be readily extended to
more complicated crystalline materials with nontrivial anharmonic lattice
effects.
