Related papers: Graph limits and exchangeable random graphs
We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$,…
For a sequence of random graphs, the limit law we refer to is the existence of a limiting probability of any graph property that can be expressed in terms of predicate logic. A zero-one limit law is shown by Shelah and Spencer for…
We show that the Aldous--Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a…
Uncertainty principles present an important theoretical tool in signal processing, as they provide limits on the time-frequency concentration of a signal. In many real-world applications the signal domain has a complicated irregular…
The celebrated Erdos, Faber and Lovasz conjecture may be stated as follows: Any linear hypergraph on v points has chromatic index at most v. We will introduce the linear intersection number of a graph, and use this number to give an…
We study some percolation problems on the complete graph over $\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the…
Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by Lov\'asz and…
Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has…
Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.
We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first…
Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius.…
The exponential family of random graphs represents an important and challenging class of network models. Despite their flexibility, conventionally used exponential random graphs have one shortcoming. They cannot directly model weighted…
We provide an upper bound to the number of graph homomorphisms from $G$ to $H$, where $H$ is a fixed graph with certain properties, and $G$ varies over all $N$-vertex, $d$-regular graphs. This result generalizes a recently resolved…
We strengthen and put in a broader perspective previous results of the first two authors on colliding permutations. The key to the present approach is a new non-asymptotic invariant for graphs.
Graph-theoretic tools and techniques have seen wide use in the multi-agent systems literature, and the unpredictable nature of some multi-agent communications has been successfully modeled using random communication graphs. Across both…
We establish a central limit theorem for the sum of $\epsilon$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting…
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…