Related papers: Randomly Generated Subgroups of the Symmetric Grou…
For a graph $G$ the random $n$-lift of $G$ is obtained by replacing each of its vertices by a set of $n$ vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of $G$ are adjacent. We show that…
Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…
An $\ell$-lift of a graph $G$ is any graph obtained by replacing every vertex of $G$ with an independent set of size $\ell$, and connecting every pair of two such independent sets that correspond to an edge in $G$ by a matching of size…
In this paper we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The…
Clique complexes of Erd\H{o}s-R\'{e}nyi random graphs with edge probability between $n^{-{1\over 3}}$ and $n^{-{1\over 2}}$ are shown to be aas not simply connected. This entails showing that a connected two dimensional simplicial complex…
Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs.…
In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…
A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…
We consider classes of pseudo-random graphs on $n$ vertices for which the degree of every vertex and the co-degree between every pair of vertices are in the intervals $(np - Cn^\delta,np+Cn^\delta)$ and $(np^2- C n^\delta, np^2 +C…
We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…
For various purposes and, in particular, in the context of data compression, a graph can be examined at three levels. Its structure can be described as the unlabeled version of the graph; then the labeling of its structure can be added; and…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
In this paper, we study the connectivity of a one-dimensional soft random geometric graph (RGG). The graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on…
Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$ with $n$ vertices, in every step $i \geq 1$ the graph $G_i$ is formed by taking an edge chosen uniformly at random among the non-existing ones and adding it to…
In the standard random graph process, edges are added to an initially empty graph one by one uniformly at random. A classic result by Ajtai, Koml\'os, and Szemer\'edi, and independently by Bollob\'as, states that in the standard random…
We suggest a new random model for links based on meander diagrams and graphs. We then prove that trivial links appear with vanishing probability in this model, no link $L$ is obtained with probability 1, and there is a lower bound for the…
Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real…
We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first…
The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of…
An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen…