Related papers: Finitely forcible graphons with an almost arbitrar…
We consider large uniform labeled random graphs in different classes with few induced $P_4$ ($P_4$ is the graph consisting of a single line of $4$ vertices) which generalize the case of cographs. Our main result is the convergence to a…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…
The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on…
We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.
We consider the problem of estimating graph limits, known as graphons, from observations of sequences of sparse finite graphs. In this paper we show a simple method that can shed light on a subset of sparse graphs. The method involves…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Exchangeable random graphs, which include some of the most widely studied network models, have emerged as the mainstay of statistical network analysis in recent years. Graphons, which are the central objects in graph limit theory, provide a…
We associate to a graphon $\gamma$ the sequence of $W$-random graphs $(G_n(\gamma))_{n \geq 1}$. We say that the graphon is singular if, for any finite graph $F$, the homomorphism density $t(F,G_n(\gamma))$ has a variance of order…
Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…
In their study of the giant component in inhomogeneous random graphs, Bollob\'as, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that the tree structures underlying two…
We define a covering of a profinite graph to be a projective limit of a system of covering maps of finite graphs. With this notion of covering, we develop a covering theory for profinite graphs which is in many ways analogous to the…
We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong…
In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons…
For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either…
We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$ such that $G\not\subseteq K$, or (ii)…
A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical…
Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and…
Graphons have traditionally served as limit objects for dense graph sequences, with the cut distance serving as the metric for convergence. However, sparse graph sequences converge to the trivial graphon under the conventional definition of…
The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and \v{Z}ivkovi\'{c}. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this…