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Abstract matrix-tree theorem [article]

Yurii Burman
2017 arXiv   pre-print
The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 relates the principal minor of the nxn Laplace matrix to a particular sum of monomials of matrix elements indexed by directed trees with  ...  In this paper we consider a generalization of this statement: for any k > n we define a degree k polynomial det_n,k of matrix elements and prove that this polynomial applied to the Laplace matrix gives  ...  Corollaries 1.9 and 1.10 are particular cases of the celebrated matrix-tree theorem first discovered by G.  ... 
arXiv:1612.03873v3 fatcat:xc4xpzb27fghto52fp54saqhku

Around matrix-tree theorem [article]

Yurii Burman, Boris Shapiro
2006 arXiv   pre-print
Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core.  ...  We use this generalization to obtain an analog of the matrix-tree theorem for the root system D_n (the classical theorem corresponds to the A_n-case).  ...  the Laplacian matrix (in the case of trees it was its principal minor).  ... 
arXiv:math/0512164v2 fatcat:2u5ep3lrljfmvf4iz2z6bg62fm

Simplicial matrix-tree theorems [article]

Art M. Duval, Caroline J. Klivans, Jeremy L. Martin
2008 arXiv   pre-print
Laplacian matrix of Δ.  ...  We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the  ...  Theorem 1.1 (Classical Matrix-Tree Theorem).  ... 
arXiv:0802.2576v2 fatcat:l4ixasu4zngpnds27g27lerngi

Simplicial matrix-tree theorems

Art M. Duval, Caroline J. Klivans, Jeremy L. Martin
2009 Transactions of the American Mathematical Society  
Laplacian matrix of ∆.  ...  We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the  ...  Introduction This article is about generalizing the Matrix-Tree Theorem from graphs to simplicial complexes. 1.1. The classical Matrix-Tree Theorem.  ... 
doi:10.1090/s0002-9947-09-04898-3 fatcat:x5rob3upqnhbfl7t5dskih753y

Higher matrix-tree theorems [article]

Yurii Burman, Andrey Ploskonosov, Anastasia Trofimova
2011 arXiv   pre-print
The results obtained generalize the Kirchhoff's matrix-tree theorem and the matrix-3-hypertree theorem by G.\,Massbaum and A.\,Vaintrob.  ...  This proves the classical matrix-tree theorem (see [3] and also [2] for another proof): Theorem 2.2 (matrix-tree theorem, [3] ).  ...  Here are two particular cases of Theorem 1.3 for the A n system of roots: 2.1. k = 1: a matrix-tree theorem.  ... 
arXiv:1109.6625v1 fatcat:zqs6ac2nrjg57deg5ot5njl2wm

Around matrix-tree theorem

Yuri Burman, Boris Shapiro
2006 Mathematical Research Letters  
Generalizing the classical matrix-tree theorem we provide a formula counting, for a given graph, its subgraphs with a fixed 2-core.  ...  We use this generalization to obtain an analog of the matrix-tree theorem for the root system Dn (the classical theorem corresponds to the An-case).  ...  the Laplacian matrix (in the case of trees it was its principal minor).  ... 
doi:10.4310/mrl.2006.v13.n5.a7 fatcat:75jx2fwwyranpdwd7b3ozetkrm

Trees with Matrix Weights: Laplacian Matrix and Characteristic-like Vertices [article]

Swetha Ganesh, Sumit Mohanty
2022 arXiv   pre-print
Furthermore, we also compute the Moore-Penrose inverse of the Laplacian matrix of a tree with nonsingular matrix weights on its edges.  ...  For trees with the above classes of matrix edge weights, we define Perron values and Perron branches.  ...  Let L be the Laplacian matrix of a tree T = (V, E) with nonsingular matrix weights on its edges.  ... 
arXiv:2009.05996v4 fatcat:axka4mzbizfwtlfhqk7iin6j6q

A New Matrix-Tree Theorem [article]

Gregor Masbaum, Arkady Vaintrob
2002 arXiv   pre-print
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix.  ...  We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably defined matrix.  ...  Theorem The Pfaffian Matrix-Tree Theorem 3.  ... 
arXiv:math/0109104v2 fatcat:v5re77m3qfby5czl4bjoaatwe4

Distance matrix polynomials of trees

R.L Graham, L Lovász
1978 Advances in Mathematics  
DISTANCE MATRIX POLYNOMIALS OF TREES 63 It is also true that h(T) = (-V-12"-9(4&I(T) + 2h2(T) + 4KJT) -4), (6) where Pz denotes the path of length 2, i.e., the unique tree with three vertices.  ...  The smallest example [A of two nonisomorphic trees having the same "spectrum," i.e., set of adjacency matrix eigenvalues, is shown in Fig. 1 . For these trees, AT,(h) = A,,(X) = X8 -7X6 + 9h4.  ... 
doi:10.1016/0001-8708(78)90005-1 fatcat:g4yfk7nc7ngx3gayckekwbshd4

Average Mixing Matrix of Trees

Chris Godsil, Krystal Guo, John Sinkovic
2018 The Electronic Journal of Linear Algebra  
A lower bound on the rank of the average mixing matrix of a tree, is also given.  ...  The rank of the average mixing matrix of trees with all eigenvalues distinct, is investigated.  ...  Average Mixing Matrix of Trees Theorem 3.4.  ... 
doi:10.13001/1081-3810.3746 fatcat:wheluqoxqrhmrgga2tvl6gf2y4

Average mixing matrix of trees [article]

Chris Godsil, Krystal Guo, John Sinkovic
2017 arXiv   pre-print
We also give a lower bound on the rank of the average mixing matrix of a tree.  ...  We investigate the rank of the average mixing matrix of trees, with all eigenvalues distinct.  ...  We then give the following lower bound on the rank of the average mixing matrix of a tree: Theorem.  ... 
arXiv:1709.07907v1 fatcat:7oaxql7tzfcghghctsxrz3bgoy

A family of matrix-tree multijections [article]

Alex McDonough
2021 arXiv   pre-print
matrix-tree theorems.  ...  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  ...  . , e r } forms a spanning tree which we call T .  ... 
arXiv:2007.09501v3 fatcat:uiddlnyvwvftpdozryh2mzlqbq

Higher determinants and the matrix-tree theorem [article]

Yurii Burman
2016 arXiv   pre-print
The classical matrix-tree theorem was discovered by G.~Kirchhoff in 1847.  ...  The aim of this paper is to present a generalization of the (nonsymmetric) matrix-tree theorem containing no trees and essentially no matrices.  ...  Introduction The classical matrix-tree theorem was discovered by G. Kirchhoff in 1847, see [1] .  ... 
arXiv:1508.02245v2 fatcat:qcoeqgkoyzbsxjsvtmdrlhca4q

Product distance matrix of a tree with matrix weights

R.B. Bapat, Sivaramakrishnan Sivasubramanian
2015 Linear Algebra and its Applications  
Let T be a tree on n vertices and let the n − 1 edges e 1 , e 2 , . . . , e n−1 have weights that are s × s matrices W 1 , W 2 , . . . , W n−1 , respectively.  ...  Define the distance between i and j as the s × s matrix E i,j = k p=1 W ep . Consider the ns × ns matrix D whose i, j-th block is the matrix E i,j .  ...  B B B B B B •5 Figure 1: A matrix-weighted tree T .  ... 
doi:10.1016/j.laa.2014.03.034 fatcat:pa4e2uqspzfpffi46r4kw6drmi

A colourful path to matrix-tree theorems [article]

Adrien Kassel, Thierry Lévy
2019 arXiv   pre-print
In this short note, we revisit Zeilberger's proof of the classical matrix-tree theorem and give a unified concise proof of variants of this theorem, some known and some new.  ...  The version of the matrix-tree theorem that we are going to state computes the τ -determinant of ∆ [m] .  ...  A matrix-tree theorem Let n 2 be an integer and let V = {1, . . . , n} be the set of vertices of a complete graph.  ... 
arXiv:1903.02491v2 fatcat:y32ajaxvi5grpitdkrtrvko6ii
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