Related papers: A note on directed analogues of the Sidorenko and …
Some families of graphs, such as the n-cubes and Sierpinski gaskets, are self-similar. In this paper we show how such recursive structure can be used systematically to prove isoperimetric theorems.
We state the analogs of Kontsevich's formality conjecture for Hochschild and cyclic chains, as well as their
Graph comparison is a certain type of condition on metric space encoded by a finite graph. We show that any nontrivial graph comparison implies one of Alexandrov's comparisons. The proof gives a complete description of graphs with trivial…
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…
A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be…
For an $r$-graph $F$, define Sidorenko exponent $s(F)$ as $$s(F):= \sup \{s \geq 0: \exists \text{$r$-graph $H$ s.t. } t_F(H) = t_{K^{(r)}_r} (H)^s > 0\},$$ where $t_{H_1}(H_2)$ denotes the homomorphism density of $H_1$ in $H_2$. The…
We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links…
In 2013 Andriy V. Bondarenko showed how to construct a two-distance counterexample to Borsuk's conjecture from any strongly regular graph whose vertex set is not the union of at most $f+1$ cliques (sets of pairwise adjacent vertices) where…
The competition graph of a directed acyclic graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The competition number of an undirected graph G is…
We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (Also conjectured by Erd\H os and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for…
Directed graphs occur throughout statistical modeling of networks, and exchangeability is a natural assumption when the ordering of vertices does not matter. There is a deep structural theory for exchangeable undirected graphs, which…
We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…
We propose a novel approach for learning node representations in directed graphs, which maintains separate views or embedding spaces for the two distinct node roles induced by the directionality of the edges. We argue that the previous…
We deal with the rigidity conjecture of symbolic powers over regular rings. This was asked by Huneke. Along with our investigation, we confirm a conjecture [7, Conjecture 3.8].
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…
Every graph G can be embedded in a Euclidean space as a two-distance set. This allows us to reformulate the analogue of Borsuk's conjecture for two-distance sets in terms of graphs. This conjecture remains open for dimensions from 4 to 63.…
We extend Andersson-Madigan-Perlman chain graphs by (i) relaxing the semidirected acyclity constraint so that only directed cycles are forbidden, and (ii) allowing up to two edges between any pair of nodes. We introduce global, and ordered…
We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.
The inductive biases of graph representation learning algorithms are often encoded in the background geometry of their embedding space. In this paper, we show that general directed graphs can be effectively represented by an embedding model…
Usually, mathematical objects have highly parallel interpretations. In this paper, we consider them as sequential constructors of other objects. In particular, we prove that every reflexive directed graph can be interpreted as a program…