Related papers: Hypergraph expanders of all uniformities from Cayl…
We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree…
We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute…
Random recursive hypergraphs grow by adding, at each step, a vertex and an edge formed by joining the new vertex to a randomly chosen existing edge. The model is parameter-free, and several characteristics of emerging hypergraphs admit neat…
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…
A $3$-uniform hypergraph is a generalization of simple graphs where each hyperedge is a subset of vertices of size $3$. The degree of a vertex in a hypergraph is the number of hyperedges incident with it. The degree sequence of a hypergraph…
We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been…
It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with…
A Cayley hyper-digraph is a directed hypergraph that its automorphism group contains a subgroup acting regularly on vertices and a Cayley hypermap is a hypermap whose automorphism group contains a subgroup which induces regular action on…
Chv\'atal, R\"odl, Szemer\'edi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an…
The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…
It is well known that 3--regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3--regular graphs without reducing the girth, thereby proving that such graphs with…
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a…
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…
This paper develops analityc methods for investigating uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues of 2-graphs. On the one hand, this simple setup is…
For each uniformity $k \geq 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a root $\lambda$ with $\lvert\lambda\rvert = O\left(\frac{\log…
We generalize the Harary-Sachs theorem to $k$-uniform hypergraphs: the codegree-$d$ coefficient of the characteristic polynomial of a uniform hypergraph ${\cal H}$ can be expressed as a weighted sum of subgraph counts over certain…
We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via $(\Delta-Y)$-transformations. We compare this…
The Cayley graphs of finite groups are known to provide several examples of families of expanders, and some of them are Ramanujan graphs. Babai studied isospectral non-isomorphic Cayley graphs of the dihedral groups. Lubotzky, Samuels and…
An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut…
Extensions of Erd\H{o}s-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erd\H{o}s-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex…