A family of matrix-tree multijections

@article{McDonough2020AFO,
  title={A family of matrix-tree multijections},
  author={Alex McDonough},
  journal={Algebraic Combinatorics},
  year={2020}
}
  • A. McDonough
  • Published 18 July 2020
  • Mathematics
  • Algebraic Combinatorics
. For a natural class of r × n integer matrices, we construct a non-convex polytope which periodically tiles R n . From this tiling, we provide a family of geometrically meaningful maps from a generalized sandpile group to a set of generalized spanning trees which give multijective proofs for several higher-dimensional matrix-tree theorems. In particular, these multijections can be ap- plied to graphs, regular matroids, cell complexes with a torsion-free spanning forest, and representable… 
View PDF on arXiv
1 Citations

Figures from this paper

One Citation

Standard monomials of 1-skeleton ideals of graphs and generalized signless Laplacians

References

SHOWING 1-10 OF 34 REFERENCES

Critical Groups of Simplicial Complexes

We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian…

GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of…

Matrix Tree Theorems

Circuit-Cocircuit Reversing Systems in Regular Matroids

Abstract.We consider that two orientations of a regular matroid are equivalent if one can be obtained from the other by successive reorientations of positive circuits and/or positive cocircuits. We…

Cuts and flows of cell complexes

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a…

Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra

Preface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville…

Tutte Polynomial, Subgraphs, Orientations and Sandpile Model: New Connections via Embeddings

The bijection $\Phi$ is closely related to a recent characterization of the Tutte polynomial relying on combinatorial embeddings of graphs, that is, on a choice of cyclic order of the edges around each vertex.

Enumerating degree sequences in digraphs and a cycle-cocycle reversing system

Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra

An algebraic framework for generalized graph Laplacians is proposed which unifies the study of resistor networks, the critical group, and the eigenvalues of theLaplacian and adjacency matrices, and an algorithm for simplifying computation of $\Upsilon_R(G,L)$ is proposed.

Algebraic Graph Theory

The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.