Related papers: Adjacency Spectra of Random and Uniform Hypergraph…

In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is…

We prove that for each $d\geq 3$ and $k\geq 2$, the set of limit points of the first $k$ eigenvalues of sequences of $d$-regular graphs is \[ \{(\mu_1,\dots,\mu_k): d=\mu_1\geq \dots\geq \mu_{k}\geq2\sqrt{d-1}\}. \] The result for $k=2$ was…

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…

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…

Much effort has been spent on characterizing the spectrum of the non-backtracking matrix of certain classes of graphs, with special emphasis on the leading eigenvalue or the second eigenvector. Much less attention has been paid to the…

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously…

It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most…

We prove that any quasirandom uniform hypergraph $H$ can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the…

This is the fourth in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this…

Let p(k) denote the partition function of k. For each k >= 2, we describe a list of p(k)-1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of…

Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large…

This paper is the second chapter of three of the author's undergraduate thesis. In this paper, we consider the random matrix ensemble given by $(d_b, d_w)$-regular graphs on $M$ black vertices and $N$ white vertices, where $d_b \in…

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory,…

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction…

Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…

We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our…

Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the…

We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that…

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as…

Let $G_1,\dots, G_m$ be independent identically distributed Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having vertex sets of random sizes $X_1,\dots, X_m\in \{0,1,2,\dots\}$ and random edge densities $Q_1,\dots, Q_m\in…