Related papers: On the very weak 0-1 law for random graphs with or…

Order the vertices of a directed random graph \math{v_1,...,v_n}; edge \math{(v_i,v_j)} for \math{i<j} exists independently with probability \math{p}. This random graph model is related to certain spreading processes on networks. We…

For any fixed positive integer $k$, let $\alpha_{k}$ denote the smallest $\alpha \in (0,1)$ such that the random graph sequence $\left\{G\left(n, n^{-\alpha}\right)\right\}$ does not satisfy the zero-one law for the set $\mathcal{E}_{k}$ of…

Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 and a_{k-1}= 1. We show under a couple of assumptions concerning the constants c_i…

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in…

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the…

Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is…

Given i.i.d. positive integer valued random variables D_1,...,D_n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D_1,...,D_n. We give sufficient conditions on the distribution of D_i for…

We consider the range of random walks up to time n, R_n, on graphs satisfying a uniform condition. This condition is characterized by potential theory. Not only all vertex transitive graphs but also many non-regular graphs satisfy the…

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly…

A graph $G$ on $n$ vertices is \textit{pancyclic} if it contains cycles of length $t$ for all $3 \leq t \leq n$. In this paper we prove that for any fixed $\epsilon>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ asymptotically almost…

In this paper, we study spectra of first order properties of Erdos-Renyi random graph. We proved that minimal quantifier depth of a formula with an infinite spectrum is either 4 or 5.

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs…

Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be the adjacency matrix of the random directed $d$-regular graph on $n$ vertices, with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with…

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 $k$…

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…

A class A of labelled graphs is bridge-addable if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and u is also in A; the class A is monotone if for all G…

We analyse the eigenvalues of Erd\"os--R\'enyi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=\Omega(\sqrt{n_{1}p}\log^{3}(n_{1})),$ $n_{2}p=\Omega(\sqrt{n_{2}p}\log^{3}(n_{2})),$ and let $G\sim…

We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of nodes representing vertices and a set of nodes representing hyper-edges, with two nodes being adjacent if the corresponding vertex belongs to the corresponding…

We consider a conditionally Poissonian random graph model where the mean degrees, `capacities', follow a power-tailed distribution with finite mean and infinite variance. Such a graph of size $N$ has a giant component which is super-small…

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all…