Independent sets in regular graphs: spectral stabilty and the hard-core model

Combinatorics Seminar
Tuesday, January 6, 2015 - 12:05
1 hour (actually 50 minutes)
Skiles 005
Emory University and University of Sao Paulo, Brazil
Let B(d,n) denote the d-regular graph on n vertices which consists of the disjoint union of complete bipartite graphs. It follows from the results of Kahn and of Zhao that among all d-regular graphs on n vertices B(d,n) maximizes the number of independent sets. In this talk, we show a spectral stability phenomenon of this result in the following sense. The eigenvalues of (the adjacency matrix) of B(d,n) are known to be d, -d and zeroes and  we show that, if the smallest eigenvalue of G is bounded away from -d, then the number of independent sets in G is exponentially smaller than that of B(d,n). Furthermore, we extend this method to study the well-known hard-core model from statistical physics. Given a d-regular bipartite graph G whose second smallest eigenvalue is bounded away from -d. Let Ind(G) denote the set of all independent sets of G. Among others, we show that in this case the random independent set I\in Ind(G), drawn from the hard-core distribution with activation parameter lambda>> (log d)/d, is essentially completely (up to o(|I|) vertices) contained in one of the partition classes of G. (This is joint work with Prasad Tetali.)