Gundert, Anna; Wagner, UliIST Austria
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 X k(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 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.
151 - 160
SGC: Symposuim on Computational Geometry
Gundert A, Wagner U. On Laplacians of random complexes. In: ACM; 2012:151-160. doi:10.1145/2261250.2261272
Gundert, A., & Wagner, U. (2012). On Laplacians of random complexes (pp. 151–160). Presented at the SGC: Symposuim on Computational Geometry, ACM. https://doi.org/10.1145/2261250.2261272
Gundert, Anna, and Uli Wagner. “On Laplacians of Random Complexes,” 151–60. ACM, 2012. https://doi.org/10.1145/2261250.2261272.
A. Gundert and U. Wagner, “On Laplacians of random complexes,” presented at the SGC: Symposuim on Computational Geometry, 2012, pp. 151–160.
Gundert A, Wagner U. 2012. On Laplacians of random complexes. SGC: Symposuim on Computational Geometry 151–160.
Gundert, Anna, and Uli Wagner. On Laplacians of Random Complexes. ACM, 2012, pp. 151–60, doi:10.1145/2261250.2261272.