February  2021, 20(2): 885-902. doi: 10.3934/cpaa.2020295

On the quotient quantum graph with respect to the regular representation

Gazi University, Faculty of Science, Department of Mathematics, Ankara-Turkey

Received  June 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

Given a quantum graph $ \Gamma $, a finite symmetry group $ G $ acting on it and a representation $ R $ of $ G $, the quotient quantum graph $ \Gamma /R $ is described and constructed in the literature [1,2,18]. In particular, it was shown that the quotient graph $ \Gamma/\mathbb{C}G $ is isospectral to $ \Gamma $ by using representation theory where $ \mathbb{C}G $ denotes the regular representation of $ G $ [18]. Further, it was conjectured that $ \Gamma $ can be obtained as a quotient $ \Gamma/\mathbb{C}G $ [18]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph $ \Gamma $ and a finite symmetry group $ G $ acting on it, the quotient quantum graph $ \Gamma / \mathbb{C}G $ is not only isospectral but rather identical to $ \Gamma $ for a particular choice of a basis for $ \mathbb{C}G $. Furthermore, we prove that, this result holds for an arbitrary permutation representation of $ G $ with degree $ |G| $, whereas it doesn't hold for a permutation representation of $ G $ with degree greater than $ |G|. $

Citation: Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295
References:
[1]

R. Band, G. Berkolaiko, C. H. Joyner and W. Liu, Quotients of finite-dimensional operators by symmetry representations, arXiv: 1711.00918v3.

[2]

R. Band, O. Parzanchevski and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor., 42 (2009), 175202. doi: 10.1088/1751-8113/42/17/175202.

[3]

R. BandT. Shapira and U. Smilansky, Nodal domains on isospectral quantum graphs: the resolution of isospectrality?, J. Phys. A: Math. Gen., 39 (2006), 13999-14014.  doi: 10.1088/0305-4470/39/45/009.

[4]

G. Berkolaiko, An elementary introduction to quantum graphs, arXiv: 1603.07356v2.

[5]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Rhode Island, 2013. doi: 10.1090/surv/186.

[6]

R. Carlson, Inverse eigenvalue problems on directed graphs, T. Am. Math. Soc., 351 (1999), 4069-4088.  doi: 10.1090/S0002-9947-99-02175-3.

[7]

W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.

[8]

S. Gnutzmann and U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, arXiv: nlin/0605028v2.

[9]

C. Gordon, P. Perry and D. Schüth, Isospectral and isoscattering manifolds: a survey of techniques and examples, in Geometry, Spectral Theory, Groups and Dynamics, Contemporary Mathematics vol. 387 (eds. M. Entov, Y. Pinchover and M. Sageev), American Mathematical Society, (2005), 157–179. doi: 10.1090/conm/387/07241.

[10]

C. GordonD. Webb and S. Wolpert, One cannot hear the shape of a drum, Bull. Am. Mat. Soc. (N.S.), 27 (1992), 134-138.  doi: 10.1090/S0273-0979-1992-00289-6.

[11]

C. GordonD. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.  doi: 10.1007/BF01231320.

[12]

B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen., 31 (2001), 6061-6068.  doi: 10.1088/0305-4470/34/31/301.

[13]

M. Kac, Can one hear the shape of a drum?, Am. Math. Mon., 73 (1966), 1-23.  doi: 10.2307/2313748.

[14]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.

[15]

T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.  doi: 10.1103/PhysRevLett.79.4794.

[16]

P. Kuchment, Quantum graphs: an introduction and a brief survey, arXiv: 0802.3442v1.

[17]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[18]

O. Parzanchevski and R. Band, Linear representations and isospectrality with boundary conditions, J. Geom. Anal., 20 (2010), 439-471.  doi: 10.1007/s12220-009-9115-6.

[19]

T. Shapira and U. Smilansky, Quantum graphs which sound the same, in Non-Linear Dynamics and Fundamental Interactions. NATO Science Series Ⅱ: Mathematics, Physics and Chemistry, vol 213. (eds. F. Khanna and D. Matrasulov), Springer, (2005), 17–29. doi: 10.1007/1-4020-3949-2_2.

[20]

T. Sunada, Riemanninan coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186.  doi: 10.2307/1971195.

[21]

J. von Below, Can one hear the shape of a network?, in Partial Differential Equations on Multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., vol. 219 (eds. F. Mehmeti, J. von Below and S. Nicaise), Dekker, (2001), 19–36.

show all references

References:
[1]

R. Band, G. Berkolaiko, C. H. Joyner and W. Liu, Quotients of finite-dimensional operators by symmetry representations, arXiv: 1711.00918v3.

[2]

R. Band, O. Parzanchevski and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor., 42 (2009), 175202. doi: 10.1088/1751-8113/42/17/175202.

[3]

R. BandT. Shapira and U. Smilansky, Nodal domains on isospectral quantum graphs: the resolution of isospectrality?, J. Phys. A: Math. Gen., 39 (2006), 13999-14014.  doi: 10.1088/0305-4470/39/45/009.

[4]

G. Berkolaiko, An elementary introduction to quantum graphs, arXiv: 1603.07356v2.

[5]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Rhode Island, 2013. doi: 10.1090/surv/186.

[6]

R. Carlson, Inverse eigenvalue problems on directed graphs, T. Am. Math. Soc., 351 (1999), 4069-4088.  doi: 10.1090/S0002-9947-99-02175-3.

[7]

W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.

[8]

S. Gnutzmann and U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, arXiv: nlin/0605028v2.

[9]

C. Gordon, P. Perry and D. Schüth, Isospectral and isoscattering manifolds: a survey of techniques and examples, in Geometry, Spectral Theory, Groups and Dynamics, Contemporary Mathematics vol. 387 (eds. M. Entov, Y. Pinchover and M. Sageev), American Mathematical Society, (2005), 157–179. doi: 10.1090/conm/387/07241.

[10]

C. GordonD. Webb and S. Wolpert, One cannot hear the shape of a drum, Bull. Am. Mat. Soc. (N.S.), 27 (1992), 134-138.  doi: 10.1090/S0273-0979-1992-00289-6.

[11]

C. GordonD. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.  doi: 10.1007/BF01231320.

[12]

B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen., 31 (2001), 6061-6068.  doi: 10.1088/0305-4470/34/31/301.

[13]

M. Kac, Can one hear the shape of a drum?, Am. Math. Mon., 73 (1966), 1-23.  doi: 10.2307/2313748.

[14]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.

[15]

T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.  doi: 10.1103/PhysRevLett.79.4794.

[16]

P. Kuchment, Quantum graphs: an introduction and a brief survey, arXiv: 0802.3442v1.

[17]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[18]

O. Parzanchevski and R. Band, Linear representations and isospectrality with boundary conditions, J. Geom. Anal., 20 (2010), 439-471.  doi: 10.1007/s12220-009-9115-6.

[19]

T. Shapira and U. Smilansky, Quantum graphs which sound the same, in Non-Linear Dynamics and Fundamental Interactions. NATO Science Series Ⅱ: Mathematics, Physics and Chemistry, vol 213. (eds. F. Khanna and D. Matrasulov), Springer, (2005), 17–29. doi: 10.1007/1-4020-3949-2_2.

[20]

T. Sunada, Riemanninan coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186.  doi: 10.2307/1971195.

[21]

J. von Below, Can one hear the shape of a network?, in Partial Differential Equations on Multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., vol. 219 (eds. F. Mehmeti, J. von Below and S. Nicaise), Dekker, (2001), 19–36.

Figure 1.  A star graph with $ C_{2} $ symmetry
Figure 2.  A star graph with 3 edges having the same length
Figure 3.  An equilateral tetrahedron graph
Figure 4.  We added dummy vertices and assigned directions arbitrarily
[1]

James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3029-3063. doi: 10.3934/cpaa.2021095

[2]

Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks and Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483

[3]

Constanza Riera, Matthew G. Parker, Pantelimon Stǎnicǎ. Quantum states associated to mixed graphs and their algebraic characterization. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021015

[4]

Sergei Avdonin, Pavel Kurasov, Marlena Nowaczyk. Inverse problems for quantum trees II: Recovering matching conditions for star graphs. Inverse Problems and Imaging, 2010, 4 (4) : 579-598. doi: 10.3934/ipi.2010.4.579

[5]

Padmapani Seneviratne, Martianus Frederic Ezerman. New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021073

[6]

Sam Northshield. Quasi-regular graphs, cogrowth, and amenability. Conference Publications, 2003, 2003 (Special) : 678-687. doi: 10.3934/proc.2003.2003.678

[7]

Joaquim Borges, Josep Rifà, Victor A. Zinoviev. Families of nested completely regular codes and distance-regular graphs. Advances in Mathematics of Communications, 2015, 9 (2) : 233-246. doi: 10.3934/amc.2015.9.233

[8]

Cristian Dobre. Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 367-378. doi: 10.3934/naco.2013.3.367

[9]

Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041

[10]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492

[11]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021

[12]

Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046

[13]

Bin Yu. Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1277-1290. doi: 10.3934/dcds.2011.29.1277

[14]

Dean Crnković, Marija Maksimović, Bernardo Gabriel Rodrigues, Sanja Rukavina. Self-orthogonal codes from the strongly regular graphs on up to 40 vertices. Advances in Mathematics of Communications, 2016, 10 (3) : 555-582. doi: 10.3934/amc.2016026

[15]

Dean Crnković, Ronan Egan, Andrea Švob. Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs. Advances in Mathematics of Communications, 2020, 14 (4) : 591-602. doi: 10.3934/amc.2020032

[16]

Dean Crnković, Sanja Rukavina, Andrea Švob. Self-orthogonal codes from equitable partitions of distance-regular graphs. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022014

[17]

Ziling Heng, Dexiang Li, Fenjin Liu, Weiqiong Wang. Infinite families of $ t $-designs and strongly regular graphs from punctured codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022043

[18]

Litao Guo, Bernard L. S. Lin. Vulnerability of super connected split graphs and bisplit graphs. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1179-1185. doi: 10.3934/dcdss.2019081

[19]

Hengrui Luo, Alice Patania, Jisu Kim, Mikael Vejdemo-Johansson. Generalized penalty for circular coordinate representation. Foundations of Data Science, 2021, 3 (4) : 729-767. doi: 10.3934/fods.2021024

[20]

Monica Pragliola, Daniela Calvetti, Erkki Somersalo. Overcomplete representation in a hierarchical Bayesian framework. Inverse Problems and Imaging, 2022, 16 (1) : 19-38. doi: 10.3934/ipi.2021039

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (229)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]