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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Shifted simplicial complexes are Laplacian integral


Authors: Art M. Duval and Victor Reiner
Journal: Trans. Amer. Math. Soc. 354 (2002), 4313-4344
MSC (2000): Primary 15A42; Secondary 05C65, 05C50, 05E99
Published electronically: July 2, 2002
MathSciNet review: 1926878
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Abstract: We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs.

We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.


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Additional Information

Art M. Duval
Affiliation: Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514
Email: artduval@math.utep.edu

Victor Reiner
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: reiner@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03082-9
PII: S 0002-9947(02)03082-9
Keywords: Laplace operator, Laplacian, simplicial complex, spectra
Received by editor(s): May 3, 2000
Received by editor(s) in revised form: August 10, 2000
Published electronically: July 2, 2002
Additional Notes: Second author partially supported by a Sloan Foundation Fellowship and NSF grant DMS-9877047.
Article copyright: © Copyright 2002 American Mathematical Society