Related papers: Size biased couplings and the spectral gap for ran…

A well known upper bound for the spectral radius of a graph, due to Hong, is that $\mu_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the…

How does the chromatic number of a graph chosen uniformly at random from all graphs on $n$ vertices behave? This quantity is a random variable, so one can ask (i) for upper and lower bounds on its typical values, and (ii) for bounds on how…

Motivated both by recently introduced forms of list colouring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi-random method to prove the following result. For any function $\mu$…

The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n…

Let $G$ be a $d$-regular multigraph, and let $\lambda_2(G)$ be the second largest eigenvalue of $G$. In this paper, we prove that if $\lambda_2(G) < \frac{d-1+\sqrt{9d^2-10d+17}}4$, then $G$ is 2-edge-connected. Furthermore, for $t\ge2$ we…

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$,…

We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if $G$ is an $n$-node weighted undirected graph of average combinatorial degree $d$ (that is, $G$ has $dn/2$ edges) and girth $g>…

Let $G$ be a connected $d$-regular graph of order $n$, where $d\geq3$. Let $\lambda_{2}(G)$ be the second largest eigenvalue of $G$. For even $n$, we show that $G$ contains $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$…

In 2004, Kim and Vu conjectured that, when $d=\omega(\log n)$, the random $d$-regular graph $G_d(n)$ can be sandwiched with high probability between two random binomial graphs $G(n,p)$ with edge probabilities asymptotically equal to…

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let $G$ be a graph on $n$ vertices. A 2-lift of $G$ is a graph $H$ on $2n$ vertices,…

We study the spectral gap of the Erd\H{o}s--R\'enyi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta > 0$ if $$p \ge \frac{(1/2 + \delta) \log n}{n},$$ then the normalized graph Laplacian of…

The on-line nearest-neighbour graph on a sequence of $n$ uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this…

Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…

For a fixed natural number $t\geq 2$, we consider $t$-uniform random hypergraphs $\mathscr{H} (n,t,p)$ on $n$ vertices $[n]=\{1,\ldots, n\}$, where each $t$-subset of $[n]$ is included as a hyperedge with probability $p$ and independently.…

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq…

We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow…

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-\Omega(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the…

This is the third 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 X_{d,n} be an n-element subset of {0,1}^d chosen uniformly at random, and denote by P_{d,n} := conv X_{d,n} its convex hull. Let D_{d,n} be the density of the graph of P_{d,n} (i.e., the number of one-dimensional faces of P_{d,n}…

An $(n,d,\lambda)$-graph is a $d$ regular graph on $n$ vertices in which the absolute value of any nontrivial eigenvalue is at most $\lambda$. For any constant $d \geq 3$, $\epsilon>0$ and all sufficiently large $n$ we show that there is a…