Related papers: Hypergraphs with Spectral Radius at most $(r-1)!\s…

We classify maximal digraphs whose Hermitian spectral radius is at most $2$.

Write $\rho\left( G\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\vee\overline{K}_{n-r}.$ Let $n>r\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined…

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the…

Let $Q$ be an $s$-vertex $r$-uniform hypergraph, and let $H$ be an $n$-vertex $r$-uniform hypergraph. Denote by $\mathcal{N}(Q,H)$ the number of isomorphic copies of $Q$ in $H$. For a hereditary family $\mathcal{P}$ of $r$-uniform…

A mixed graph is a graph with undirected and directed edges. Guo and Mohar in 2017 determined all mixed graphs whose Hermitian spectral radii are less than $2$. In this paper, we give a sufficient condition which can make Hermitian spectral…

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph $G$ with $\rho(G)\leq 3/2{\sqrt{2}}$ is either a dagger, an open quipu, or a closed quipu.…

For $0\leq \alpha < 1$, the $\mathcal{A}_{\alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_{\alpha}(G):=\alpha \mathcal{D}(G)+(1-\alpha) \mathcal{A}(G)$, where…

This paper investigates the maximum spectral radius of planar graphs with concrete fixed number of vertices, providing some tight bounds on the maximum spectral radius of general planar graph resorting to its order, and confirming that…

Research on the relationship of the (signless Laplacian) spectral radius of a graph with its structure properties is an important research project in spectral graph theory. Denote by $\rho(G)$ and $q(G)$ the spectral radius and the signless…

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 $\tau(\mathcal{H})$ be the cover number and $\nu(\mathcal{H})$ be the matching number of a hypergraph $\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\mathcal{H}$ satisfies the inequality $\tau(\mathcal{H}) \leq…

The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are often tighter and are applicable to a larger class of digraphs than previously reported…

Erd\H{o}s and Lov\'asz noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We…

Let $\mathcal{H}^{(r)}_n$ be the set of all connected $r$-graphs with given size $n$. In this paper, we investigate the effect on the spectral radius of $r$-uniform hypergraphs by grafting or contracting an edge and then give the ordering…

For a $hypergraph$ $\mathcal{G}=(V, E)$ consisting of a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as…

Let $\mathbb{Q}_{k,n}$ be the set of the connected $k$-uniform weighted hypergraphs with $n$ vertices, where $k,n\geq 3$. For a hypergraph $G\in \mathbb{Q}_{k,n}$, let $\mathcal{A}(G)$, $\mathcal{L} (G)$ and $\mathcal{Q} (G)$ be its…

Let $G$ be an $r$-uniform hypergraph of order $t$ and $\rho(G)$ is the spectral radius of $\mathcal{A}(G)$, where $\mathcal{A}(G)$ is the adjacency tensor of $G$. A blow-up of $G$ respected to a positive integer vector $(n_{1},…

The $p$-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a…

Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the…

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be…