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On laplacians of random complexes.
| Content Provider | CiteSeerX |
|---|---|
| Author | Gundert, Anna Wagner, Uli |
| Abstract | 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) 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 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 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 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 k>1, 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. |
| File Format | |
| Access Restriction | Open |
| Subject Keyword | Random Complex Edge Expansion Large Eigenvalue Gap Simplicial Complex Second Part Higher-dimensional Analogue Close Relationship Fundamental Inequality Cheeger Inequality Well-studied Subject K-dimensional Simplicial Complex Possible Higherdimensional Analogue Higher-dimensional Cheeger Inequality Linial Meshulam Model Adjacency Matrix Combinatorial Expansion Property Higher-dimensional Adjacency Matrix Random Graph K-dimensional Complex Higher-dimensional Version Spectral Expansion Discrete Cheeger Inequality Straightforward Version Infinite Family |
| Content Type | Text |
