# American Institute of Mathematical Sciences

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

## Nodal geometry of graphs on surfaces

 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 Davieset. 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 graphsatisfying a certain volume-growth condition, the multiplicity ofthe $n$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.
Citation: Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291
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