Related papers: Arithmetic expanders and deviation bounds for rand…
For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$. We focus on algebraic vectors, namely $\overline…
Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of…
The Cayley sum graph $\Gamma_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$…
We study the appearance of the giant component in random subgraphs of a given large finite graph G=(V,E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then…
This is the third in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this…
We compute an asymptotic expansion in $1/c$ of the limit in $n$ of the empirical spectral measure of the adjacency matrix of an Erd\H{o}s-R\'enyi random graph with $n$ vertices and parameter $c/n$. We present two different methods, one of…
A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq…
We confirm the long-standing prediction that $c=e\approx 2.718$ is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erd\H{o}s-Renyi random graph with…
We establish a large deviation principle (LDP) for probability graphons, which are symmetric functions from the unit square into the space of probability measures. This notion extends classical graphons and provides a flexible framework for…
The $r$-expansion of a $k$-uniform hypergraph $H$, denoted by $H^{(+r)}$, is an $r$-uniform hypergraph obtained by enlarging each $k$-edge of $H$ with a set of $r-k$ vertices of degree one. The random Tur\'an number…
The Eulerian extension number of any graph~\(H\) (i.e. the minimum number of edges needed to be added to make~\(H\) Eulerian) is at least~\(t(H),\) half the number of odd degree vertices of~\(H.\) In this paper we consider an inhomogenous…
The on-line nearest-neighbour graph on a sequence of $n$ uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this…
We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular…
In a seminal paper, Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of $G$…
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic…
We investigate the following vertex percolation process. Starting with a random regular graph of constant degree, delete each vertex independently with probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We show…
The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several…
We prove a $pre$-$asymptotic$ bound on the total variation distance between the uniform distribution over two types of undirected graphs with $n$ nodes. One distribution places a prescribed number of $k_T$ triangles and $k_S$ edges not…
We establish an Expander Mixing Lemma for directed graphs in terms of the eigenvalues of an associated asymmetric transition probability matrix, extending the classical spectral inequality to the asymmetric setting. As an application, we…
Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be…