Combinatorica
On the second eigenvalue of random regular graphs
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
On the second eigenvalue of a graph
Discrete Mathematics
On the Betti Numbers of Chessboard Complexes
Journal of Algebraic Combinatorics: An International Journal
Spectral techniques applied to sparse random graphs
Random Structures & Algorithms
Quasirandomness, Counting and Regularity for 3-Uniform Hypergraphs
Combinatorics, Probability and Computing
Homological Connectivity Of Random 2-Complexes
Combinatorica
On the Laplacian Eigenvalues of Gn,p
Combinatorics, Probability and Computing
An elementary construction of constant-degree expanders
Combinatorics, Probability and Computing
Homological connectivity of random k-dimensional complexes
Random Structures & Algorithms
Minors in random and expanding hypergraphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem
Discrete & Computational Geometry
High dimensional expanders and property testing
Proceedings of the 5th conference on Innovations in theoretical computer science
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Eigenvalues associated to graphs are a well-studied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n,p) are known quite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues for the Linial--Meshulam model Xk(n,p) of random k-dimensional simplicial complexes on n vertices. We show that for p=Ω(log n/n), the eigenvalues of both, the higher-dimensional 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 higher-dimensional 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 higher-dimensional 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 higher-dimensional version of Cheeger's inequality. We show that the most straightforward version of a higher-dimensional Cheeger inequality fails: for every k1, there is an infinite family of k-dimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.