[1] P. Kuchment, Quantum graphs: I. Some basic structures, Waves Random Media 14, S107 (2004).
[2] P. Kuchment, Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A 38, 4887 (2005).
[3] V. Kostrykin and R. Schrader, Kirchoff’s rule for quantum wires, J. Phys. A 32, 595 (1999).
[4] C. Texier and G. Montambaux, Scattering theory on graphs, J. Phys. A 34, 10307 (2001).
[5] J. Boman and P. Kurasov, Symmetries of quantum graphs and the inverse scattering problem, Adv. Appl. Math. 35, 58 (2005).
[6] Y. Fujimoto, K. Konno, T. Nagasawa, and R. Takahashi, Quantum reflection and transmission in ring systems with double Y-junctions: Occurrence of perfect reflection, J. Phys. A 53, 155302 (2020).
[7] T. Kottos and U. Smilansky, Quantum Chaos on Graphs, Phys. Rev. Lett. 79, 4794 (1997).
[8] P. Hejčík and T. Cheon, Irregular dynamics in a solvable one-dimensional quantum graph, Phys. Lett. A 356, 290 (2006).
[9] ík and T. Cheon, Irregular dynamics in a solvable one-dimensional quantum graph, Phys. Lett. A 356, 290 (2006).
[10] S. Gnutzmann, J. Keating, and F. Piotet, Eigenfunction statistics on quantum graphs, Ann. Phys. (Amsterdam) 325, 2595 (2010).
[11] J. M. Harrison, J. P. Keating, J. M. Robbins, and A. Sawicki, n-particle quantum statistics on graphs, Commun. Math. Phys. 330, 1293 (2014).
[12] T. Maciążek and A. Sawicki, Homology groups for particles on one-connected graphs, J. Math. Phys. (N.Y.) 58, 062103 (2017).
[13] T. Maciążek and A. Sawicki, Non-Abelian quantum sta- tistics on graphs, Commun. Math. Phys. 371, 921 (2019).
[14] T. Nagasawa, M. Sakamoto, and K. Takenaga, Supersym- metry and discrete transformations on S1 with point singu- larities, Phys. Lett. B 583, 357 (2004).
[15] T. Nagasawa, M. Sakamoto, and K. Takenaga, Extended supersymmetry and its reduction on a circle with point singularities, J. Phys. A 38, 8053 (2005).
[16] S. Ohya, Parasupersymmetry in Quantum Graphs, Ann. Phys. (Amsterdam) 331, 299 (2013).
[17] S. Ohya, Non-Abelian monopole in the parameter space of point-like interactions, Ann. Phys. (Amsterdam) 351, 900 (2014).
[18] S. Ohya, BPS monopole in the space of boundary con- ditions, J. Phys. A 48, 505401 (2015).
[19] S. Ohya, Models for the BPS Berry connection, Mod. Phys. Lett. A 36, 2150007 (2021).
[20] T. Inoue, M. Sakamoto, and I. Ueba, Instantons and Berry’s connections on quantum graph, J. Phys. A 54, 355301 (2021).
[21] H. D. Kim, Hiding an extra dimension, J. High Energy Phys. 01 (2006) 090.
[22] G. Cacciapaglia, C. Csaki, C. Grojean, and J. Terning, Field theory on multi-throat backgrounds, Phys. Rev. D 74, 045019 (2006).
[23] A. Bechinger and G. Seidl, Resonant Dirac leptogenesis on throats, Phys. Rev. D 81, 065015 (2010).
[24] S. Abel and J. Barnard, Strong coupling, discrete symmetry and flavour, J. High Energy Phys. 08 (2010) 039.
[25] S. S. C. Law and K. L. McDonald, Broken symmetry as a stabilizing remnant, Phys. Rev. D 82, 104032 (2010).
[26] Y. Fujimoto, T. Nagasawa, K. Nishiwaki, and M. Sakamoto, Quark mass hierarchy and mixing via geometry of extra dimension with point interactions, Prog. Theor. Exp. Phys. 2013, 023B07 (2013).
[27] Y. Fujimoto, K. Nishiwaki, and M. Sakamoto, CP phase from twisted Higgs vacuum expectation value in extra dimension, Phys. Rev. D 88, 115007 (2013).
[28] Y. Fujimoto, K. Nishiwaki, M. Sakamoto, and R. Takahashi, Realization of lepton masses and mixing angles from point interactions in an extra dimension, J. High Energy Phys. 10 (2014) 191.
[29] Y. Fujimoto, T. Miura, K. Nishiwaki, and M. Sakamoto, Dynamical generation of fermion mass hierarchy in an extra dimension, Phys. Rev. D 97, 115039 (2018).
[30] Y. Fujimoto, K. Hasegawa, K. Nishiwaki, M. Sakamoto, K. Takenaga, P. H. Tanaka, and I. Ueba, Dynamical generation of quark/lepton mass hierarchy in an extra dimension, Prog. Theor. Exp. Phys. 2019, 123B02 (2019).
[31] F. Nortier, Large star/rose extra dimension with small leaves/petals, Int. J. Mod. Phys. A 35, 2050182 (2020).
[32] Y. Fujimoto, T. Inoue, M. Sakamoto, K. Takenaga, and I. Ueba, 5D Dirac fermion on quantum graph, J. Phys. A 52, 455401 (2019).
[33] A. Schnyder, S. Ryu, A. Furusaki, and A. Ludwig, Clas- sification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
[34] A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009).
[35] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
[36] G. Y. Cho, K. Shiozaki, S. Ryu, and A. W. W. Ludwig, Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum, J. Phys. A 50, 304002 (2017).
[37] B. Han, A. Tiwari, C.-T. Hsieh, and S. Ryu, Boundary conformal field theory and symmetry protected topological phases in 2 1 dimensions, Phys. Rev. B 96, 125105 (2017).
[38] D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
[39] M. R. Zirnbauer, Riemannian symmetric superspaces and their origin in random-matrix theory, J. Math. Phys. (N.Y.) 37, 4986 (1996).
[40] A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid struc- tures, Phys. Rev. B 55, 1142 (1997).
[41] T. Kugo and P. K. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B221, 357 (1983).
[42] C. G. Callan, Jr. and J. A. Harvey, Anomalies and fermion zero modes on strings and domain walls, Nucl. Phys. B250, 427 (1985).
[43] E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88, 035001 (2016).
[44] E. Witten and K. Yonekura, Anomaly inflow and the η- invariant, in Proceedings of the Shoucheng Zhang Memorial Workshop (Stanford, CA, 2019), p. 9, arXiv:1909.08775.
[45] L. Fidkowski and A. Kitaev, The effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81, 134509 (2010).
[46] L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83, 075103 (2011).
[47] S. Ryu and S.-C. Zhang, Interacting topological phases and modular invariance, Phys. Rev. B 85, 245132 (2012).
[48] L. Fidkowski, X. Chen, and A. Vishwanath, Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model, Phys. Rev. X 3, 041016 (2013).
[49] C. Wang and T. Senthil, Interacting fermionic topological insulators/superconductors in three dimensions, Phys. Rev. B 89, 195124 (2014); 91, 239902(E) (2015).
[50] Y.-Z. You and C. Xu, Symmetry protected topological states of interacting fermions and bosons, Phys. Rev. B 90, 245120 (2014).
[51] T. Morimoto, A. Furusaki, and C. Mudry, Breakdown of the topological classification Z for gapped phases of noninter- acting fermions by quartic interactions, Phys. Rev. B 92, 125104 (2015).
[52] A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, J. High Energy Phys. 12 (2015) 052.
[53] S. Ryu, J. E. Moore, and A. W. W. Ludwig, Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors, Phys. Rev. B 85, 045104 (2012).
[54] X.-G. Wen, Classifying gauge anomalies through sym- metry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys. Rev. D 88, 045013 (2013).
[55] C.-T. Hsieh, G. Y. Cho, and S. Ryu, Global anomalies on the surface of fermionic symmetry-protected topological phases in (3 1) dimensions, Phys. Rev. B 93, 075135 (2016).
[56] E. Witten, The parity anomaly on an unorientable manifold, Phys. Rev. B 94, 195150 (2016).