Related papers: On the largest eigenvalue of a sparse random subgr…

We analyse the largest eigenvalue of the adjacency matrix of the configuration model with large degrees, where the latter are treated as hard constraints. In particular, we compute the expectation of the largest eigenvalue for degrees that…

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of…

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…

It is known that the complete graph $K_n$ contains a pancyclic subgraph with $n+(1+o(1))\cdot \log _2 n$ edges, and that there is no pancyclic graph on $n$ vertices with fewer than $n+\log _2 (n-1) -1$ edges. We show that, with high…

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c…

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an…

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…

Let d \geq d_0 be a sufficiently large constant. A (n,d,c \sqrt{d}) graph G is a d-regular graph over n vertices whose second largest (in absolute value) eigenvalue is at most c \sqrt{d}. For any 0 < p < 1, G_p is the graph induced by…

We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order \[ \Omega\left( \sqrt{\frac{n…

Let $\mathcal A$ be the adjacency matrix of the Erd\H{o}s-R\'{e}nyi directed graph $\mathscr G(N,p)$. We denote the eigenvalues of $\mathcal A$ by $\lambda_1^{\cal A},...,\lambda^{\cal A}_N$, and $|\lambda_1^{\cal A}|=\max_i|\lambda_i^{\cal…

It is proved that the median eigenvalues of every connected bipartite graph $G$ of maximum degree at most three belong to the interval $[-1,1]$ with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$.…

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for…

We consider the binomial random graph $G(n,p)$, where $p$ is a constant, and answer the following two questions. First, given $e(k)=p{k\choose 2}+O(k)$, what is the maximum $k$ such that a.a.s.~the binomial random graph $G(n,p)$ has an…

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a…

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected…

We give an upper bound on the smallest eigenvalue of the adjacency matrix of graphs with no p-cliques.

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We prove that the…

In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erd\H{o}s-R\'enyi random graph $\mathcal{G}_{n,p}$, in the regime of $p$ where the edge of the spectrum is no…

$Q_{n,p}$, the random subgraph of the $n$-vertex hypercube $Q_n$, is obtained by independently retaining each edge of $Q_n$ with probability $p$. We give precise values for the cover time of $Q_{n,p}$ above the connectivity threshold.

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of $k$-regular graphs. We also prove an analogue of Serre's theorem regarding the least eigenvalues of $k$-regular graphs: given…