On Kirchhoff’s theorems with coefficients in a line bundle

Catanzaro, Michael J.; Chernyak, Vladimir; Klein, John R.

doi:10.4310/hha.2013.v15.n2.a16

Cited by 5 publications

(9 citation statements)

References 8 publications

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“…From this point of view the above combinatorial results are in the setting of the trivial U(1) representation. As also done for some results on spanning trees by M. Catanzaro, V. Chernyak, and J. Klein [5], in this section the above combinatorial results are extended to U(1) representations.…”

Section: Variant Settings and Examplesmentioning

confidence: 85%

“…The papers of M. Catanzaro, V. Chernyak, and J. Klein, [5,5,6] also study aspects of the Reidemeister-Fraz torion in the combinatorial context. Their results are complementary to those given here.…”

Section: Combinatorial Identities Via a Computation Of Reidemeister-fmentioning

confidence: 99%

“…Of course, there are analogous theorems about forests and coforests and their dual notions. As mentioned above, the spanning tree case of coefficients in a flat complex line bundle where k = dim B d−1 (X; E ρ ), is discussed in M. Catanzaro, V. Chernyak, and J. Klein [5] and there related directly to circuit theory. 11.3.…”

Section: Variant Settings and Examplesmentioning

confidence: 99%

“…These definitions and combinatorial identities appear in §10. A use of Reidemeister-Franz torsion in the combinatorial context also appears in the work of Catanzaro, Chernyak, Klein [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Enumerative Combinatorics of Simplicial and Cell Complexes: Kirchhoff and Trent Type Theorems

Cappell

Miller

2018

Discrete Comput Geom

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This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of subobjects. The matrices are: the mesh matrix for integral d-cycles, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral (d-1)-chains.Trent's theorem states that the determinant of the mesh matrix on 1-cycles of a connected graph is equal to the number of spanning trees [29,30]. Here this theorem is extended to the mesh matrix on d cycles in an arbitrary cell complex and, new even for graphs, to enumerative combinatorial interpretation of all of the coefficients of its characteristic polynomial. This last is well defined once a basis for the integral d-cycles is chosen. Additionally, a parallel result for the mesh matrix for integral d-boundaries is proved.Kirchhoff's theorem states that the product of the non-zero eigenvalues of the Kirchhoff matrix, i.e., combinatorial Laplacian, for connected graphs equals n times the number of spanning trees with n the number of vertices. Lyons has generalized Kirchhoff's result on the product of the non-zero eigenvalues of the Kirchhoff or combinatorial Laplacian on (d-1)-chains to cell complexes for d > 1 [18]. The present analysis extends this to all coefficients of the characteristic polynomial.An evaluation of the Reidemeister-Franz torsion of the cell complex with respect to its integral basis gives relations between these combinatorial invariants. Section 1. Homology covolume squared via determinants of mesh matrices Section 10. Combinatorial identities via a computation of Reidemeister-Franz torsion Section 11. Weights, Variant Settings, and Examples Section 12. Enumerative combinatorics of the simplex ( extending Kalai [14] )

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Section: Variant Settings and Examplesmentioning

confidence: 85%

Section: Combinatorial Identities Via a Computation Of Reidemeister-fmentioning

confidence: 99%

Section: Variant Settings and Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Enumerative Combinatorics of Simplicial and Cell Complexes: Kirchhoff and Trent Type Theorems

Cappell

Miller

2018

Discrete Comput Geom

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“…Connection Laplacian methods have proven enlightening in the study of synchronization problems. Others have approached the study of vector bundles, and in particular line bundles, over graphs without reference to the connection Laplacian, studying analogues of spanning trees and the Kirchhoff theorems [Ken11,CCK13]. Other work on discrete approximations to connection Laplacians of manifolds has analyzed similar matrices [Man07].…”

Section: Comparison With Previous Constructionsmentioning

confidence: 99%

Toward a spectral theory of cellular sheaves

Hansen

Ghrist

2019

J Appl. and Comput. Topology

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This paper outlines a program in what one might call spectral sheaf theory -an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes results on eigenvalue interlacing, complex sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.

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