## 4 Theory of the Thermal Conductivity of Graphene and GNR  
The first experimental investigations of the thermal properties in graphene materials [4, 5, 31, 32, 33, 37] stimulated numerous theoretical and computational works in the field (see Figure 2a,b). Here, we briefly review the state-of-the-art in theory of thermal transport in graphene and GNRs. Many different theoretical models have been proposed for the prediction of the phonon and thermal properties in graphite, graphene and GNRs during the last few years. The phonon energy spectra have been theoretically investigated using Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) [38, 39, 40], valence-force-field (VFF) and Born-von Karman models of lattice vibrations [41, 42, 43, 44, 45, 46], continuum approach [47, 48, 49], first-order local density approximation [39, 50, 51], fifth- and fourth-nearest neighbor force constant approaches [40, 52] or utilized the Tersoff, Brenner or Lennard-Jones potentials [53, 54, 55]. The thermal conductivity investigations have been performed within molecular dynamics simulations [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72], density functional theory [73, 74], [PERSON]'s function method [75, 76] and Boltzmann-transport-equation (BTE) approach [31, 41, 42, 43, 49, 53, 54, 55, 56, 77, 78, 79, 80, 81, 82, 83, 84, 85]. It has been shown that phonon energies strongly depend on the interatomic force constants (IFCs)--fitting parameters of interatomic interactions, used in the majority of the models. Therefore a proper choice of interatomic force constants is crucial for the accurate description of phonon energy spectra and thermal conductivity in graphene, twisted graphene and graphene nanoribbons [1, 2, 3, 44, 86]. Although various models predicted different values of thermal conductivity, they demonstrated consistent results on the strong dependence of graphene lattice thermal conductivity on the extrinsic parameters of flakes: edge quality, FLG thickness, lateral size and shape, lattice strain, isotope, impurity and grain concentration. The molecular dynamic (MD) simulations usually give smaller values of thermal conductivity in comparison with the BTE model and experimental data due to exclusion of long wavelength phonons from the model by a finite size of the simulation domain [2]. The effect of the edge roughness on the thermal conductivity in graphene and GNRs has been investigated in Refs. [41, 42, 43, 49, 57, 79, 80, 81, 82, 84, 85, 87, 88]. The rough edge can suppress the thermal conductivity by an order of magnitude as compared to that in graphene or GNRs with perfect edges due to the boundary scattering of phonons. Impurities, single vacancies, double vacancies and Stone-Wales defects decrease the thermal conductivity of graphene and GNRs by more than 50%-80% in dependence of the defect concentration [41, 42, 43, 63, 64, 65, 66, 67, 68, 69, 79]. A study of thermal conductivity of graphene and GNRs under strain was performed in Refs. [61, 73, 74, 76, 89]. An enhancement of the thermal conductivity of up to 36% for the strained 5-nm armchair or zigzag GNRs was found in the ballistic transport regime [76]. In the diffusive transport regime, the applied strain enhanced the Umklapp scattering and thermal conductivity diminishes by \(\sim\)1.4 orders of magnitude at RT in comparison with the unstrained graphene [74]. The authors of Ref. [61] have found that when the strain is applied in both directions--parallel and perpendicular to the heat transfer path--the graphene sheets undergo complex reconstructions. As a result, some of the strained graphene structures can have higher thermal conductivity than that of SLG without strain [61]. The discrepancy between theoretical findings and experiments requires additional investigations of thermal transport in strained graphene and GNRs. The isotope composition is another key parameter for thermal conductivity engineering in these materials [1; 2; 3; 83; 90; 91; 92; 93; 94; 95].