Regular slices for hypergraphs

We present a ‘Regular Slice Lemma’ which, given a k -graph GG, returns a regular (k−1)(k−1)-complex JJ with respect to which GG has useful regularity properties. We believe that many arguments in extremal hypergraph theory are made considerably simpler by using this lemma rather than existing forms of the Strong Hypergraph Regularity Lemma, and advocate its use for this reason

The minimum vertex degree for an almost-spanning tight cycle in a 33-uniform hypergraph

We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1411.495

Combinatorial methods for the spectral p-norm of hypermatrices

The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of $2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of $2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to $r$-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for $2$-matrices. The main result of this paper extends the latter result to $r$-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on combinatorial concepts like $r$-partite $r$-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral $p$-norm in general. Thus, another application shows that the spectral $p$-norm and the $p$-spectral radius of a symmetric nonnegative $r$-matrix are equal whenever $p\geq r$. This result contributes to a classical area of analysis, initiated by Mazur and Orlicz around 1930. Additionally, a number of bounds are given on the $p$-spectral radius and the spectral $p$-norm of $r$-matrices and $r$-graphs.Comment: 29 pages. Credit has been given to Ragnarsson and Van Loan for the symmetrant of a matri

Tight cycles in hypergraphs

We apply a recent version of the Strong Hypergraph Regularity Lemma(see [1], [2]) to prove two new results on tight cycles in k-uniform hypergraphs. The first result is an extension of the Erdos-Gallai Theorem for graphs: For every > 0, every sufficiently large k-uniform hypergraph on n vertices with at least edges contains a tight cycle of length @n for any @ 2 [0; 1]. Our second result concerns k-partite k-uniform hypergraphs with partition classes of size n and for each @ 2 (0; 1) provides an asymptotically optimal minimum codegree requirement for the hypergraph to contain a cycle of length @kn

Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics

Hypergraphs and graph products extend traditional graph theory by incorporating multi-way and coupled relationships, which are ubiquitous in real-world systems. While the Kronecker product, rooted in matrix analysis, has become a powerful tool in network science, its application has been limited to pairwise networks. In this paper, we extend the coupling of graph products to hypergraphs, enabling a system-theoretic analysis of network compositions formed via the Kronecker product of hypergraphs. We first extend the notion of the matrix Kronecker product to the tensor Kronecker product from the perspective of tensor blocks. We present various algebraic and spectral properties and express different tensor decompositions with the tensor Kronecker product. Furthermore, we study the structure and dynamics of Kronecker hypergraphs based on the tensor Kronecker product. We establish conditions that enable the analysis of the trajectory and stability of a hypergraph dynamical system by examining the dynamics of its factor hypergraphs. Finally, we demonstrate the numerical advantage of this framework for computing various tensor decompositions and spectral properties.Comment: 29 pages, 4 figures, 2 table