Related papers: Sharp lower bounds on the spectral radius of unifo…

Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of $G$, respectively. The number of components of $G$ is denoted by $c(G)$. Let $t$ be a positive real number, and a connected graph $G$ is…

Let $\mathcal{H}$ be a hypergraph on a finite set $V$. A {\em cover} of $\mathcal{H}$ is a set of vertices that meets all edges of $\mathcal{H}$. If $W$ is not a cover of $\mathcal{H}$, then $W$ is said to be a {\em noncover} of…

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For an $r$-uniform graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…

Let $a$, $b$, and $n$ be three integers such that $1\leq a \leq b < n$, $a \equiv b$ (mod $2$), and $na$ is even. A parity $[a,b]$-factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $a \leq d_H(v) \leq b$ and…

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…

In this paper, we give the relationship between spectral radius and local structures of graphs and hypergraphs. Our work shows that certain local subgraphs (subhypergraphs) must occur when the spectral radius ratio is large. We also give…

For an $r$-uniform hypergraph $G$, let $\lambda^{(p)}(G)$ denote its $p$-spectral radius, defined as the maximum of the polyform of $G$ over the unit sphere in the $\ell_p$-norm. Let $Q_k^r(n)$ be the complete $k$-chromatic $r$-graph on $n$…

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of…

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a $k$-uniform hypergraph property with sensitivity complexity $O(n^{\lceil k/3\rceil})$ for any $k\geq3$, where $n$ is the number of vertices.…

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of…

Let $n$ and $r$ be integers with $n-2\ge r\ge 3$. We prove that any $r$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with spectral radius $\lambda(\mathcal{H}) > \binom{n-2}{r-1}$ must contain a Hamiltonian Berge cycle unless…

We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data. Let H be such a hypergraph on a set of vertices V. We give a lower bound of…

Let $G$ be a $k$-uniform hypergraph with vertex set $V(G)$ and edge set $E(G)$. A connected and acyclic hypergraph is called a supertree. For $0\leq\alpha<1$, the $\alpha$-spectral radius of $G$ is the largest $H$-eigenvalue of $\alpha…

Bollob\'as, Reed and Thomason proved every $3$-uniform hypergraph $\mathcal{H}$ with $m$ edges has a vertex-partition $V(\mathcal{H})=V_1 \sqcup V_2 \sqcup V_3$ such that each part meets at least $\frac{1}{3}(1-\frac{1}{e})m$ edges, later…

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if…

Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral…

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as \[ \lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{…

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian…

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with…

We study the problem of testing the existence of a dense subhypergraph. The null hypothesis is an Erdos-Renyi uniform random hypergraph and the alternative hypothesis is a uniform random hypergraph that contains a dense subhypergraph. We…