Related papers: Explicit laws of large numbers for random nearest-…

The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with…

We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al.~\cite{ar:Krioukov} as a model for complex networks, $N$ vertices are drawn randomly within a bounded subset of the hyperbolic…

Delaunay and Gabriel graphs are widely studied geometric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as \emph{Locally Delaunay Graphs} ($LDGs$) and \emph{Locally Gabriel…

In this note, we introduce a family of bipartite graphs called path restricted ordered bipartite graphs and present it as an abstract generalization of some well known geometric graphs like unit distance graphs on convex point sets. In the…

In this paper, we asymptotically enumerate graphs with a given degree sequence d=(d_1,...,d_n) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general…

This is an extended version of the thesis presented to the Programa de P\'os-Gradua\c{c}\~ao em Matem\'atica of the Departamento de Matem\'atica, PUC-Rio, in September 2013, incorporating some suggestions from the examining commission.…

Nearest neighbor search and k-nearest neighbor graph construction are two fundamental issues arise from many disciplines such as multimedia information retrieval, data-mining and machine learning. They become more and more imminent given…

Motivated by a conjecture of Gy\'arf\'as, recently B\"ottcher, Hladk\'y, Piguet, and Taraz showed that every collection $T_1,\dots,T_t$ of trees on $n$ vertices with $\sum_{i=1}^te(T_i)\leq \binom{n}{2}$ and with bounded maximum degree, can…

We prove new explicit inapproximability results for the Vertex Cover Problem on the Power Law Graphs and some functional generalizations of that class of graphs. Our results depend on special bounded degree amplifier constructions for those…

Building upon the theory of graph limits and the Aldous-Hoover representation and inspired by Panchenko's work on asymptotic Gibbs measures (Annals of Probability 2013), we construct continuous embeddings of discrete probability…

Let H = (V,E) be a k-uniform hypergraph with a vertex set V and an edge set E. Let V_p be constructed by taking every vertex in V independently with probability p. Let X be the number of edges in E that are contained in V_p. We give a…

We define a random graph obtained via connecting each point of $\mathbb{Z}^d$ independently to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed $k$-neighbor graph. Two natural…

Consider a stationary Poisson process $\eta$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $\eta$ as follows. First, each point $x\in\eta$ is connected by an edge to its nearest neighbour,…

Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in…

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions…

We derive the distribution of the maximum number of common neighbours of a pair of vertices in a dense random regular graph.The proof involves two important steps. One step is to establish the extremal independence property: the asymptotic…

Inspired by previous work of Diaz, Petit, Serna, and Trevisan (Approximating layout problems on random graphs Discrete Mathematics, 235, 2001, 245--253), we show that several well-known graph layout problems are approximable to within a…

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u \in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is…

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a \emph{seed} node and is allowed to make queries about neighbours of nodes…

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…