ABSTRACT In our previous work  we calculated RKKY interaction between two magnetic impurities in pristine graphene using the Green’s functions (GF) in the coordinate-imaginary time representation. Now we show that the calculations of the GF in this representation can be simplified by using the Feynman’s trick, which allows to easily calculate RKKY interaction in gapped graphene. We also present calculations of the RKKY interaction in gapped or doped graphene using the coordinate-imaginary frequency representation. Both representations, corresponding to calculation of the bubble diagram in Euclidean space, have an important advantage over those corresponding to calculation in Minkowskii space, which are very briefly reviewed in the Appendix to the present work. The former, in distinction to the latter, operate only with the convergent integrals from the start to the end of the calculation.
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E. Kogan, "RKKY Interaction in Gapped or Doped Graphene," Graphene, Vol. 2 No. 1, 2013, pp. 8-12. doi: 10.4236/graphene.2013.21002.
 E. Kogan, “RKKY Interaction in Graphene,” Physical Review B, Vol. 84, No. 11, 2011, Article ID: 115119.
 M. A. H. Vozmediano, M. P. Lopez-Sancho, T. Stauber and F. Guinea, “Local Defects and Ferromagnetism in Graphene Layers,” Physical Review B, Vol. 72, No. 15, 2005, Article ID: 155121.
 V. K. Dugaev, V. I. Litvinov and J. Barnas, “Exchange Interaction of Magnetic Impurities in Graphene,” Physi- cal Review B, Vol. 74, No. 22, 2006, Article ID: 224438.
 L. Brey, H. A. Fertig and S. D. Sarma, “Diluted Graphene Antiferromagnet,” Physical Review Letters, Vol. 99, No. 11, 2007, Article ID: 116802.
 S. Saremi, “RKKY in Half-Filled Bipartite Lattices: Graphene as an Example,” Physical Review B, Vol. 76, No. 18, 2007, Article ID: 184430.
 V. V. Cheianov, O. Syljuasen, B. L. Altshuler and V. Fal’ko, “Ordered States of Adatoms on Graphene,” Phy- sical Review B, Vol. 80, No. 23, 2009, Article ID: 233409.
 A. M. Black-Schaffer, “RKKY Coupling in Graphene,” Physical Review B, Vol. 81, No. 20, 2010, pp. Article ID: 205416. doi:10.1103/PhysRevB.81.205416
 M. Sherafati and S. Satpathy, “RKKY Interaction in Graphene from the Lattice Green’s function,” Physical Review B, Vol. 83, No. 16, 2011, Article ID: 165425.
 B. Uchoa, T. G. Rappoport and A. H. Castro Neto, “Kon- do Quantum Criticality of Magnetic Adatoms in Gra- phene,” Physical Review Letters, Vol. 106, No. 1, 2011, Article ID: 016801.
 J. E. Bunder and H.-H. Lin, “Ruderman-Kittel-Kasuya- Yosida Interactions on a Bipartite Lattice,” Physical Re- view B, Vol. 80, No. 15, 2009, Article ID: 153414.
 S. R. Power and M. S. Ferreira, “Electronic Structure of Graphene beyond the Linear Dispersion Regime,” Physi- cal Review B, Vol. 83, No. 15, 2011, Article ID: 155432.
 M. Sherafati and S. Satpathy, “Analytical Expression for the RKKY Interaction in Doped Graphene,” Physical Re- view B, Vol. 84, No. 12, 2011, Article ID: 125416.
 K. Szalowski, “RKKY Coupling between Impurity Spins in Graphene Nanoflakes,” Physical Review B, Vol. 84, No. 20, 2011, Article ID: 205409.
 J. E. Bunder and J. M. Hill, “Geometric Influence on Ruderman-Kittel-Kasuya-Yosida Interactions in Zigzag Carbon Nanotubes,” Journal of Chemical Physics, Vol. 136, No. 15, 2012, Article ID: 154504.
 H. Lee, J. Kim, E. R. Mucciolo, G. Bouzerar and S. Ket- temann, “RKKY Interaction in Disordered Graphene,” Physical Review B, Vol. 85, No. 7, 2012, Article ID: 075420. doi:10.1103/PhysRevB.85.075420
 M. Sherafati and S. Satpathy, “On the Ruderman-Kittel- Kasuya-Yosida Interaction in Graphene,” Proceedings of the International Workshop on Functional Materials, Berhampur, 20-22 December 2011, pp. 24-33.
 L. Jiang, X. Lu, W. Gao, G. Yu, Z. Liu and Y. Zheng, “RKKY Interaction in AB-Stacked Multilayer Graphene,” Journal of Physics: Condensed Matter, Vol. 24, No. 20, 2012, Article ID: 206003.
 J. Klinovaja and D. Loss, “RKKY Interaction in Carbon Nanotubes and Graphene Nanoribbons.”
 O. Roslyak, G. Gumbs and D. Huang, “Gap-Modulated Doping Effects on Indirect Exchange Interaction between Magnetic Impurities in Graphene.” arXiv:1211.5099v1.
 M. I. Katsnelson, “Graphene: Carbon in Two Dimen- sions,” Cambridge University Press, Cambridge, 2012.
 R. P. Feynman, “Quantum Electrodynamics,” Westview Press, 1961.
 A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, “Integrals and Series,” Gordon and Breach Science Publishers, 1986.
 V. M. Pereira, J. M. B. Lopes dos Santos and A. H. Castro Neto, “ Modeling Disorder in Graphene,” Physical Review B, Vol. 77, No. 11, 2008, Article ID: 115109.
 A. A. Abrikosov, L. P. Gorkov and I. E. Dzyloshinski, “Methods of Quantum Field Theory in Statistical Phys- ics,” Pergamon Press, 1965.
 V. K. Dugaev, V. I. Litvinov and P. P. Petrov, “Magnetic Impurity Interactions in a Quantum Well on the Base of IV-VI Semiconductors,” Superlattices and Microstruc- tures, Vol. 16, No. 4, 1994, pp. 413-417 .
 B. Fischer and M. W. Klein, “Magnetic and Nonmagnetic Impurities in Two-Dimensional Metals,” Physical Review B, Vol. 11, No. 5, 1975, pp. 2025-2029.
 V. I. Litvinov and V. K. Dugaev, “RKKY Interaction in One- and Two-Dimensional Electron Gases,” Physical Review B, Vol. 58, No. 7, 1998, pp. 3584-3585.