Journal of Combinatorics

Volume 6 (2015)

Number 3

Pseudodeterminants and perfect square spanning tree counts

Pages: 295 – 325

DOI: http://dx.doi.org/10.4310/JOC.2015.v6.n3.a3

Authors

Jeremy L. Martin (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)

Molly Maxwell (Flathead Valley Community College, Kalispell, Montana, U.S.A.)

Victor Reiner (School of Mathematics, University of Minnesota, Minneapolis, Mn., U.S.A.)

Scott O. Wilson (Department of Mathematics, Queens College, Queens, New York, U.S.A.)

Abstract

The pseudodeterminant $\mathrm{pdet}(M)$ of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If $\partial$ is a symmetric or skew-symmetric matrix then $\mathrm{pdet}({\partial \partial}^t) = \mathrm{pdet}(\partial)^2$. Whenever $\partial$ is the $k^\textrm{th}$ boundary map of a self-dual CW-complex $X$, this linear-algebraic identity implies that the torsion-weighted generating function for cellular $k$-trees in $X$ is a perfect square. In the case that $X$ is an antipodally self-dual CW-sphere of odd dimension, the pseudodeterminant of its $k^\textrm{th}$ cellular boundary map can be interpreted directly as a torsion-weighted generating function both for $k$-trees and for $(k-1)$-trees, complementing the analogous result for even-dimensional spheres given by the second author. The argument relies on the topological fact that any self-dual even-dimensional CW-ball can be oriented so that its middle boundary map is skew-symmetric.

Keywords

pseudodeterminant, spanning tree, Laplacian, Dirac operator, perfect square, central reflex, self-dual

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