Factorizations of some weighted spanning tree enumerators

We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors.Comment: Final version, 12 pages. To appear in the Journal of Combinatorial Theory, Series A. The paper has been reorganized, and the proof of Theorem 4 shortened, in light of a more general result appearing in reference [6

Pseudodeterminants and perfect square spanning tree counts

The pseudodeterminant $\textrm{pdet}(M)$ 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 $\textrm{pdet}(\partial\partial^t)=\textrm{pdet}(\partial)^2$. Whenever $\partial$ is the $k^{th}$ 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 \emph{antipodally} self-dual CW-sphere of odd dimension, the pseudodeterminant of its $k$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.Comment: Final version; minor revisions. To appear in Journal of Combinatoric

Cyclotomic and simplicial matroids

This is the author's accepted manuscript.We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid µn represented by then th roots of unity 1, ζ, ζ2, …, ζn-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of µn in the case that n has two prime factors

Factorizations of some weighted spanning tree enumerators

This is the author's accepted manuscript.We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors

The CPWF Working Paper series: Proposed new directions and notes for Contributors

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Mechanical Coupling in Myosin V: A Simulation Study

Myosin motor function depends on the interaction between different domains that transmit information from one part of the molecule to another. The interdomain coupling in myosin V is studied with restrained targeted molecular dynamics using an all-atom representation in explicit solvent. To elucidate the origin of the conformational change due to the binding of ATP, targeting forces are applied to small sets of atoms (the forcing sets, FSs) in the direction of their displacement from the rigor conformation, which has a closed actin-binding cleft, to the post-rigor conformation, in which the cleft is open. The “minimal” FS that results in extensive structural changes in the overall myosin conformation is composed of ATP, switch 1, and the nearby HF, HG, and HH helices. Addition of switch 2 to the FS is required to achieve a complete opening of the actin-binding cleft. The restrained targeted molecular dynamics simulations reveal the mechanical coupling pathways between (i) the nucleotide-binding pocket (NBP) and the actin-binding cleft, (ii) the NBP and the converter, and (iii) the actin-binding cleft and the converter. Closing of the NBP due to ATP binding is tightly coupled to the opening of the cleft and leads to the rupture of a key hydrogen bond (F441N/A684O) between switch 2 and the SH1 helix. The actin-binding cleft may mediate the rupture of this bond via a connection between the HW helix, the relay helix, and switch 2. The findings are consistent with experimental studies and a recent normal mode analysis. The present method is expected to be useful more generally in studies of interdomain coupling in proteins

Rigidity theory for matroids

This is the author's accepted manuscript.Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in Rd in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M. The ideas of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this wider setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial), and have little to do with the special nature of graphic matroids or the field R. Our main result is a “nesting theorem” relating the various kinds of independence. Immediate corollaries include generalizations of Laman's Theorem, as well as the equality of 2-rigidity and 2-parallel independence. A key tool in our study is the space of photos of M, a natural algebraic variety whose irreducibility is closely related to the notions of rigidity independence and parallel independence. The number of points on this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation