Enumerative Combinatorics of Simplicial and Cell Complexes

Kirchhoff and Trent Type Theorems

Sylvain Cappell, Edward Miller

Research output: Contribution to journalArticle

Abstract

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 sub-objects. 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 (Trent in Proc Nat Acad Sci USA 40:1004–1007, 1954; J Acoust Soc Am 27(3):500–527, 1955). 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 classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497–508, 1847) 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. Kalai (Isr J Math 45(4): 337–351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. 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 (Lyons in J Topol Anal 1(2), 153–175, 2009; Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete 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.

Original languageEnglish (US)
JournalDiscrete and Computational Geometry
DOIs
StateAccepted/In press - Jan 1 2018

Fingerprint

Enumerative Combinatorics
Cell Complex
Simplicial Complex
Theorem
Mesh
Characteristic polynomial
Spanning tree
Cycle
Polynomials
Torsional stress
Torsion
Connected graph
Coefficient
Graph in graph theory
Combinatorial Enumeration
Eigenvalue
Homology Groups
Weighting
Well-defined
Counting

Keywords

  • Combinatorial Laplacian
  • Complexes
  • Enumerative combinatorics
  • Geometrical combinatorics
  • High dimensional generalizations of graph theory
  • Kirchhhoff’s theorem
  • Reidemeister–Franz torsion
  • Simplicial complexes
  • Spanning trees
  • Trent’s theorem

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

@article{0907216d03f543f69e04fd9cc16f90a0,
title = "Enumerative Combinatorics of Simplicial and Cell Complexes: Kirchhoff and Trent Type Theorems",
abstract = "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 sub-objects. 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 (Trent in Proc Nat Acad Sci USA 40:1004–1007, 1954; J Acoust Soc Am 27(3):500–527, 1955). 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 classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497–508, 1847) 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. Kalai (Isr J Math 45(4): 337–351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. 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 (Lyons in J Topol Anal 1(2), 153–175, 2009; Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete 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.",
keywords = "Combinatorial Laplacian, Complexes, Enumerative combinatorics, Geometrical combinatorics, High dimensional generalizations of graph theory, Kirchhhoff’s theorem, Reidemeister–Franz torsion, Simplicial complexes, Spanning trees, Trent’s theorem",
author = "Sylvain Cappell and Edward Miller",
year = "2018",
month = "1",
day = "1",
doi = "10.1007/s00454-018-0041-x",
language = "English (US)",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",

}

TY - JOUR

T1 - Enumerative Combinatorics of Simplicial and Cell Complexes

T2 - Kirchhoff and Trent Type Theorems

AU - Cappell, Sylvain

AU - Miller, Edward

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 sub-objects. 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 (Trent in Proc Nat Acad Sci USA 40:1004–1007, 1954; J Acoust Soc Am 27(3):500–527, 1955). 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 classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497–508, 1847) 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. Kalai (Isr J Math 45(4): 337–351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. 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 (Lyons in J Topol Anal 1(2), 153–175, 2009; Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete 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.

AB - 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 sub-objects. 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 (Trent in Proc Nat Acad Sci USA 40:1004–1007, 1954; J Acoust Soc Am 27(3):500–527, 1955). 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 classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497–508, 1847) 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. Kalai (Isr J Math 45(4): 337–351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. 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 (Lyons in J Topol Anal 1(2), 153–175, 2009; Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete 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.

KW - Combinatorial Laplacian

KW - Complexes

KW - Enumerative combinatorics

KW - Geometrical combinatorics

KW - High dimensional generalizations of graph theory

KW - Kirchhhoff’s theorem

KW - Reidemeister–Franz torsion

KW - Simplicial complexes

KW - Spanning trees

KW - Trent’s theorem

UR - http://www.scopus.com/inward/record.url?scp=85057330275&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057330275&partnerID=8YFLogxK

U2 - 10.1007/s00454-018-0041-x

DO - 10.1007/s00454-018-0041-x

M3 - Article

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -