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On Laplacians of Random Complexes
"... Eigenvalues associated to graphs are a wellstudied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n, p) areknownquite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues ..."
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Eigenvalues associated to graphs are a wellstudied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n, p) areknownquite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues for the Linial–Meshulam model X k (n, p) ofrandom kdimensional simplicial complexes on n vertices. We show that for p = Ω(log n/n), the eigenvalues of both, the higherdimensional adjacency matrix and the Laplacian, are a.a.s. sharply concentrated around two values. In a second part of the paper, we discuss a possible higherdimensional analogue of the Discrete Cheeger Inequality. This fundamental inequality expresses a close relationship between the eigenvalues of a graph and its combinatorial expansion properties; in particular, spectral expansion (a large eigenvalue gap) implies edge expansion. Recently, a higherdimensional analogue of edge expansion for simplicial complexes was introduced by Gromov, and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is natural to ask whether there is a higherdimensional version of Cheeger’s inequality. We show that the most straightforward version of a higherdimensional Cheeger inequality fails: for every k>1, there is an infinite family of kdimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.
Critical groups of simplicial complex
 Annals of Combinatorics
"... Abstract. 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 operators, generalizing the construction of the critical group of a graph. We show how to rea ..."
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Abstract. 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 operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups. Résumé. Nous généralisons la théorie des groupes critiques des graphes aux complexes simpliciaux. Plus précisément, pour un complexe simplicial, nous définissons une famille de groupes abéliens en termes d'opérateurs de Laplace combinatoires, qui généralise la construction du groupe critique d'un graphe. Nous montrons comment réaliser ces groupes critiques explicitement comme conoyaux des opérateurs de Laplace réduits combinatoires, et montrons qu'ils sont finis. Leurs ordres sont obtenus en comptant (avec des poids) des arbres simpliciaux couvrants. Nous décrivons comment les groupes critiques d'un complexe représentent le flux le long de ses faces, et esquissons une autre interprétation potentielle comme analogues des groupes de Chow.
Random complexes and ℓ 2 Betti numbers
, 2005
"... Abstract. Uniform spanning trees on finite graphs and their analogues on infinite graphs are a wellstudied area. On a Cayley graph of a group, we show that they are related to the first ℓ 2Betti number of the group. Our main aim, however, is to present the basic elements of a higherdimensional an ..."
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Cited by 7 (4 self)
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Abstract. Uniform spanning trees on finite graphs and their analogues on infinite graphs are a wellstudied area. On a Cayley graph of a group, we show that they are related to the first ℓ 2Betti number of the group. Our main aim, however, is to present the basic elements of a higherdimensional analogue on finite and infinite CWcomplexes, which relate to the higher ℓ 2Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin. §1. Introduction. Enumeration of spanning trees in graphs began with Kirchhoff (1847). Cayley (1889) evaluated this number in the special case of a complete graph. Cayley’s theorem was
CELLULAR SPANNING TREES AND LAPLACIANS OF CUBICAL COMPLEXES
, 2009
"... We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigen ..."
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Cited by 6 (3 self)
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We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin’s enumeration of spanning trees of a complete colorful simplicial complex from the cellular MatrixTree Theorem together with a result of Kook, Reiner and Stanton.
GENERALIZED LOOPERASED RANDOM WALKS AND APPROXIMATE REACHABILITY
"... In this paper we extend the looperased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson’s algorithm for uniform sampling of spanning trees to directed hypergraphs. In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial ..."
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In this paper we extend the looperased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson’s algorithm for uniform sampling of spanning trees to directed hypergraphs. In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial time. Our main application is to the reachability problem, also known as the directed allterminal network reliability problem. This classical problem is known to be #Pcomplete, hence is most likely intractable [BP2]. We show that in the case of bidirected graphs, a conjectured polynomial bound for the expected running time of the generalized Wilson algorithm implies a FPRAS for approximating reachability.
A polynomial invariant and duality for triangulations
 Electronic J. Combin
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