Nodal geometry of graphs on surfaces

Pages: 1291 - 1298, Volume 28, Issue 3, November 2010

doi:10.3934/dcds.2010.28.1291       Abstract        Full Text (131.4K)       Related Articles

Yong Lin - Department of Mathematics, Harvard University, Cambridge, MA 02138, United States (email)
Gábor Lippner - Department of Mathematics, Harvard University, Cambridge, MA 02138, United States (email)
Dan Mangoubi - Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel (email)
Shing-Tung Yau - Department of Mathematics, Harvard University, Cambridge, MA 02138, United States (email)

Abstract: 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:  Nodal domain, multiplicity of eigenvalues, genus.test
Mathematics Subject Classification:  Primary: 39A12; Secondary: 05C10, 35P05.

Received: April 2010;      Published: April 2010.