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