Related papers: Bounded quantifier depth spectrum for random unifo…

We analyse uniformly random proper $k$-colourings of sparse graphs with maximum degree $\Delta$ in the regime $\Delta < k\ln k $. This regime corresponds to the lower side of the shattering threshold for random graph colouring, a…

Let $n, r, k$ be positive integers such that $3\leq k < n$ and $2\leq r \leq k-1$. Let $m(n, r, k)$ denote the maximum number of edges an $r$-uniform hypergraph on $n$ vertices can have under the condition that any collection of $i$ edges,…

Let $b(k,\ell,\theta)$ be the maximum number of vertices of valency $k$ in a $(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $\theta$. We obtain an upper bound for $b(k,\ell,\theta)$ for $0 < \theta < \sqrt{k-1} +…

Let $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies…

Let $H$ be a fixed $k$-vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an $n$-vertex graph contains $H$ as a…

The work deals with the threshold probablity for r-colorability in the binomial model H(n,k,p) of a random k-uniform hypergraph. We prove a lower bound for this threshold which improves the previously known results in the wide range of the…

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit…

The study of upper density problems on Ramsey theory was initiated by Erd\H{o}s and Galvin in 1993. In this paper we are concerned with the following problem: given a fixed finite graph $F$, what is the largest value of $\lambda$ such that…

Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given…

We study the structural constraint of random scale-free networks that determines possible combinations of the degree exponent $\gamma$ and the upper cutoff $k_c$ in the thermodynamic limit. We employ the framework of graphicality…

Let $\mathcal{H}$ be a uniform hypergraph. Let $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ be the adjacency tensor and the signless Laplacian tensor of $\mathcal{H}$, respectively. In this note we prove several bounds for the spectral radius of…

For $r\geq 2$ and $p\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$\rho_p(H)=\max_{{\bf x}\in \mathbb{R}^n: \|{\bf x}\|_p=1}r \cdot \!\!\!\! \sum_{\{i_1,i_2,\ldots, i_r\}\in E(H)}…

Let $K\_{[k,t]}$ be the complete graph on $k$ vertices from which a set of edges, induced by a clique of order $t$, has been dropped. In this note we give two explicit upper bounds for $R(K\_{[k\_1,t\_1]},\dots, K\_{[k\_r,t\_r]})$ (the…

We show that on graphs with n vertices, the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n^2/log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n^2) is…

We introduce an algorithm that constructs a random uniform graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

General Berry-Esseen bounds are developed for the exponential distribution using Stein's method. As an application, a sharp error term is obtained for Hora's result that the spectrum of the Bernoulli-Laplace Markov chain has an exponential…

We consider a model for complex networks that was recently proposed as a model for complex networks by Krioukov et al. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at…

The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have…

We give variants of the Krein bound and the absolute bound for graphs with a spectrum similar to that of a strongly regular graph. In particular, we investigate what we call approximately strongly regular graphs. We apply our results to…