The paper investigates expansion properties of the Grassmann graph, motivated by recent results of [KMS16, DKK16] concerning hardness of the Vertex-Cover and of the 2-to-1 Games problems. Proving the hypotheses put forward by these papers seems to first require a better understanding of these expansion properties. We consider the edge expansion of small sets, which is the probability of choosin...

A new segmentation algorithm exploits local image quantities which are invariant to changing illumination. Local object-background probability estimates are obtained by comparing illumination invariant quantities in an actual image with the corresponding quantities in a reference image. The objects' simply connectedness is included directly into the probability estimates and leads to an iterati...

We investigate the gap formation probability of the effective one dimensional gas model recently proposed for the energy level statistics for disordered solids at the mobility edge. It is found that in order to get the correct form for the gap probability of this model, the thermodynamic limit must be taken very carefully. † [email protected] †† [email protected] To appear in the In...

A new segmentation algorithm exploits local image quantities which are invariant to changing illumination. Local object-background probability estimates are obtained by comparing illumination invariant quantities in an actual image with the corresponding quantities in a reference image. The objects' simply connectedness is included directly into the probability estimates and leads to an iterati...

Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the ` n 2 ́ undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show ...

Let G be a finite graph with minimum degree r. Form a random subgraph Gp of G by taking each edge of G into Gp independently and with probability p. We prove that for any constant ǫ > 0, if p = 1+ǫ r , then Gp is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs. AMS Classification: 05C80, 05C10.

Percolation is essentially the study of what happens to a graph when one chooses to remove vertices or edges at random. In particular, one usually chooses some probability, q, and then chooses to remove each edge (or vertex) independently with probability q. We typically study the size and shape of the largest connected component in the remaining graph. We will only study the case in which edge...

We investigate the gap formation probability of the effective one dimensional gas model recently proposed for the energy level statistics for disordered solids at the mobility edge. It is found that in order to get the correct form for the gap probability of this model, the thermodynamic limit must be taken very carefully. † [email protected] †† [email protected] To appear in the In...

Letpc{d) be the critical probability for percolation in Z . It is shown that lim,, _, ro 2dpc(d) = 1. The proof uses the properties of a random subgraph of an m-ary d-dimensional cube. If each edge in this cube is included with probability greater than \/2d{\ — 3/m), then, for large d, the cube will have a connected component of size cm for some c> 0. This generalizes a result of Ajtai, Komlds ...

In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $delta ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we ...