Related papers: Adjacency Spectra of Random and Uniform Hypergraph…

We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bi-directional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around…

Let $\mathcal A$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $\lambda_1\geq \lambda_2\cdots \geq \lambda_{N}$. For $N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of…

Spectral correlations in unitary invariant, non-Gaussian ensembles of large random matrices possessing an eigenvalue gap are studied within the framework of the orthogonal polynomial technique. Both local and global characteristics of…

We extend the results of Zhang et al. to show that $\lambda$ is an eigenvalue of a $k$-uniform hypertree $(k \geq 3)$ if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to…

We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk…

We consider the adjacency matrices of sparse random graphs from the Chung-Lu model, where edges are added independently between the $N$ vertices with varying probabilities $p_{ij}$. The rank of the matrix $(p_{ij})$ is some fixed positive…

In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996). Our main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs. We then relate the…

Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add…

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a},…

In tensor eigenvalue problems, one is likely to be more interested in H-eigenvalues of tensors. The largest H-eigenvalue of a nonnegative tensor or of a uniform hypergraph is the spectral radius of the tensor or of the uniform hypergraph.…

The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple…

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…

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…

In this paper we investigate Ramanujan hypergraphs by using hypergraph coverings. We first show that the spectrum of a $k$-fold covering $\bar{H}$ of a connected hypergraph $H$ contains the spectrum of $H$, and that it is the union of the…

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95\% of all finite subgraphs, and the…

Let $\lambda^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $\Delta$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow…

We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first…

We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices…

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for…

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all…