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Shifted simplicial complexes are Laplacian integral
Author(s):
Art
M.
Duval;
Victor
Reiner
Journal:
Trans. Amer. Math. Soc.
354
(2002),
4313-4344.
MSC (2000):
Primary 15A42;
Secondary 05C65, 05C50, 05E99
Posted:
July 2, 2002
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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:
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
Posted:
July 2, 2002
Additional Notes:
Second author partially supported by a Sloan Foundation Fellowship and NSF grant DMS-9877047.
Copyright of article:
Copyright
2002,
American Mathematical Society
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