Spectral bounds for vanishing of cohomology and the neighborhood complex of a random graph
Samir Shukla, Dhandapani Yogeshwaran · arXiv (Cornell University) · 2018
In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds lead to generalizations of the result of Aharoni, Berger and Meshulam (2005), which is valid only for clique complexes. As an application, we decrease by a logarithmic factor, the upper bound for the threshold for vanishing of cohomology of the neighborhood complex of the Erdos-Renyi random graph derived by Kahle (2007). We also increase the lower bound for the above threshold by a polynomial factor.