Shifted simplicial complexes are Laplacian integral

Art M. Duval, Victor Reiner

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)4313-4344
Number of pages32
JournalTransactions of the American Mathematical Society
Issue number11
StatePublished - Nov 2002


  • Laplace operator
  • Laplacian
  • Simplicial complex
  • Spectra


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