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 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

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Homotopy groups of Hom complexes of graphs

The notion of $\times$-homotopy from \cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space \Hom_*(G,H) with the homotopy groups of \Hom_*(G,H^I). Here \Hom_*(G,H) is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual \Hom complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that \pi_i \big(\Hom_*(G,H) \big) \cong [G,\Omega^i H]_{\times}, where $\Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{\times}$ is the set of $\times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of \cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.Comment: 20 pages, 6 figures, final version, to be published in J. Combin. Theory Ser.