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This paper presents an atomistic Green’s function study of phonon transport through a heterogeneous interface between bulk TiC substrates and graphene nanoribbons (GNRs). The force constants that govern the lattice dynamical equations are obtained from first-principles density functional theory (DFT) calculations and then optimized for the Green’s function formulation. Phonon vibrational properties of TiC and GNRs are investigated by lattice dynamics calculations with optimized force constants that correlate well to direct DFT results. Thermal conductances of TiC-GNR-TiC systems are studied together with TiC-GNR structures. The conductances of TiC-GNR interfaces are normalized by ribbon width and are found to converge. The converged value is used to estimate the interface resistance of multiwalled carbon nanotubes (MWCNTs) grown on metal catalyst support substrates and is found to be consistent in an order of magnitude sense with experimental results on MWCNT arrays. The results reveal that covalent bonds may be formed during CNT synthesis and quantify the resulting thermal impedance caused by phonon mismatch.