Related papers: Applications of the Harary-Sachs Theorem for Hyper…
Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for…
In this paper we asymptotically count $d$-regular $k$-uniform hypergraphs on $n$ vertices, provided $k$ is fixed and $d=d(n)=o(n^{1/2})$. In doing so, we extend to hypergraphs a switching technique of McKay and Wormald.
We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is…
A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary…
We study both $H$ and $E/Z$-eigenvalues of the adjacency tensor of a uniform multi-hypergraph and give conditions for which the largest positive $H$ or $Z$-eigenvalue corresponds to a strictly positive eigenvector. We also investigate when…
In this paper we develop a framework to study observability for uniform hypergraphs. Hypergraphs, being extensions of graphs, allow edges to connect multiple nodes and unambiguously represent multi-way relationships which are ubiquitous in…
An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have…
Spectral hypergraph theory studies the structural properties of a hypergraph that can be inferred from the eigenvalues and the eigenvectors of either matrices or tensors associated with it. In this paper we study the spectral…
The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal…
We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical…
Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex…
In this paper extremal problems for uniform hypergraphs are studied in the general setting of hereditary properties. It turns out that extremal problems about edges are particular cases of a general analyic problem about a recently…
We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erd\H{o}s-R\'enyi graph is cospectral when the average…
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger…
A $h$-sunflower in a hypergraph is a family of edges with $h$ vertices in common. We show that if we colour the edges of a complete hypergraph in such a way that any monochromatic $h$-sunflower has at most $\lambda$ petals, then it contains…
Gy\'arf\'as famously showed that in every $r$-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second…
The Hosoya polynomial of a graph encompasses many of its metric properties, for instance the Wiener index (alias average distance) and the hyper-Wiener index. An expression is obtained that reduces the computation of the Hosoya polynomials…
The Kruskal--Katona theorem determines the maximum number of $d$-cliques in an $n$-edge $(d-1)$-uniform hypergraph. A generalization of the theorem was proposed by Bollob\'as and Eccles, called the partial shadow problem. The problem asks…
The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this…
We study off-diagonal Ramsey numbers $r(H, K_n^{(k)})$ of $k$-uniform hypergraphs, where $H$ is a fixed linear $k$-uniform hypergraph and $K_n^{(k)}$ is complete on $n$ vertices. Recently, Conlon et al.\ disproved the folklore conjecture…