American Institute of Mathematical Sciences

Advanced
July  2010, 28(3): 1291-1298. doi: 10.3934/dcds.2010.28.1291

Nodal geometry of graphs on surfaces

Yong Lin 1, , Gábor Lippner 1, , Dan Mangoubi 2, and  Shing-Tung Yau 1,

1. 

Department of Mathematics, Harvard University, Cambridge, MA 02138, United States, United States, United States
2. 

Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel

Received  April 2010 Published  April 2010

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the $n$th Laplacian eigenvalue is at most $2[ 6(n-1) + 15(2g-2)]^2$. Our results hold for any Schrödinger operator, not just the Laplacian.
Keywords: genus.test, multiplicity of eigenvalues, Nodal domain.
Mathematics Subject Classification: Primary: 39A12; Secondary: 05C10, 35P0.
Citation: Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291
[1]

Fábio R. Pereira. Multiplicity results for fractional systems crossing high eigenvalues. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2069-2088. doi: 10.3934/cpaa.2017102
[2]

Peter Poláčik. On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2657-2667. doi: 10.3934/dcds.2014.34.2657
[3]

Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5153-5166. doi: 10.3934/dcds.2013.33.5153
[4]

Alvaro Sandroni, Eran Shmaya. A prequential test for exchangeable theories. Journal of Dynamics & Games, 2014, 1 (3) : 497-505. doi: 10.3934/jdg.2014.1.497
[5]

Li Li, Xinzhen Zhang, Zheng-Hai Huang, Liqun Qi. Test of copositive tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 881-891. doi: 10.3934/jimo.2018075
[6]

Sébastien Gautier, Lubomir Gavrilov, Iliya D. Iliev. Perturbations of quadratic centers of genus one. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 511-535. doi: 10.3934/dcds.2009.25.511
[7]

Peter K. Friz, I. Kukavica, James C. Robinson. Nodal parametrisation of analytic attractors. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 643-657. doi: 10.3934/dcds.2001.7.643
[8]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13
[9]

Josep M. Miret, Jordi Pujolàs, Nicolas Thériault. Trisection for supersingular genus $2$ curves in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 375-387. doi: 10.3934/amc.2014.8.375
[10]

Steve Limburg, David Grant, Mahesh K. Varanasi. Higher genus universally decodable matrices (UDMG). Advances in Mathematics of Communications, 2014, 8 (3) : 257-270. doi: 10.3934/amc.2014.8.257
[11]

Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617
[12]

Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003
[13]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39
[14]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
[15]

Nuno Franco, Luís Silva. Genus and braid index associated to sequences of renormalizable Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 565-586. doi: 10.3934/dcds.2012.32.565
[16]

Stefan Erickson, Michael J. Jacobson, Jr., Andreas Stein. Explicit formulas for real hyperelliptic curves of genus 2 in affine representation. Advances in Mathematics of Communications, 2011, 5 (4) : 623-666. doi: 10.3934/amc.2011.5.623
[17]

Peter Birkner, Nicolas Thériault. Efficient halving for genus 3 curves over binary fields. Advances in Mathematics of Communications, 2010, 4 (1) : 23-47. doi: 10.3934/amc.2010.4.23
[18]

Monica Lazzo, Paul G. Schmidt. Nodal properties of radial solutions for a class of polyharmonic equations. Conference Publications, 2007, 2007 (Special) : 634-643. doi: 10.3934/proc.2007.2007.634
[19]

Sarah Day; William D. Kalies; Konstantin Mischaikow and Thomas Wanner. Probabilistic and numerical validation of homology computations for nodal domains. Electronic Research Announcements, 2007, 13: 60-73.

[20]

Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences