The neighborhood complex of a random graph

For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovasz that if N[G] is k-connected, then the chromatic number of G is at least k + 3. We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(log d), compared to the expected dimension d of the complex itself.Comment: 9 pages; stated theorems more clearly and slightly generalized, and fixed one or two typo

Topology of random simplicial complexes: a survey

This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure

Topology of random clique complexes

In a seminal paper, Erdos and Renyi identified the threshold for connectivity of the random graph G(n,p). In particular, they showed that if p >> log(n)/n then G(n,p) is almost always connected, and if p << log(n)/n then G(n,p) is almost always disconnected, as n goes to infinity. The clique complex X(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erdos-Renyi Theorem. We study here the higher homology groups of X(G(n,p)). For k > 0 we show the following. If p = n^alpha, with alpha - 1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if -1/k < alpha < -1/(k+1), then it is almost always nonvanishing. We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of spheres.Comment: 23 pages; final version, to appear in Discrete Mathematics. At suggestion of anonymous referee, a section briefly summarizing the topological prerequisites has been added to make the article accessible to a wider audienc

Coboundary expanders

We describe a natural topological generalization of edge expansion for graphs to regular CW complexes and prove that this property holds with high probability for certain random complexes.Comment: Version 2: significant rewrite. 18 pages, title changed, and main theorem extended to more general random complexe