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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Simplicial matrix-tree theorems
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by Art M. Duval, Caroline J. Klivans and Jeremy L. Martin PDF
Trans. Amer. Math. Soc. 361 (2009), 6073-6114 Request permission

Abstract:

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.
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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
  • Caroline J. Klivans
  • Affiliation: Departments of Mathematics and Computer Science, The University of Chicago, Chicago, Illinois 60637
  • MR Author ID: 754274
  • Jeremy L. Martin
  • Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66047
  • MR Author ID: 717661
  • Received by editor(s): February 27, 2008
  • Published electronically: June 15, 2009
  • Additional Notes: The second author was partially supported by NSF VIGRE grant DMS-0502215
    The third author was partially supported by an NSA Young Investigators Grant
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 6073-6114
  • MSC (2000): Primary 05A15; Secondary 05E99, 05C05, 05C50, 15A18, 57M15
  • DOI: https://doi.org/10.1090/S0002-9947-09-04898-3
  • MathSciNet review: 2529925