Related papers: Annulus graphs in $\mathbb R^d$

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…

A metric graph is a pair $(G,d)$, where $G$ is a graph and $d:E(G) \to\mathbb{R}_{\geq0}$ is a distance function. Let $p \in [1,\infty]$ be fixed. An isometric embedding of the metric graph $(G,d)$ in $\ell_p^k = (\mathbb{R}^k, d_p)$ is a…

A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius…

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. This is a natural extension of the study of regular graphs, and of the study of graphs of…

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e \in \Gamma(v)} f(e) \le 1$ for each $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The {\it fractional…

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$,…

We give essentially tight bounds for, $\nu(d,k)$, the maximum number of distinct neighbourhoods on a set $X$ of $k$ vertices in a graph with twin-width at most~$d$. Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet…

In this paper, we provide a characterization of uniquely representable two-directional orthogonal ray graphs, which are defined as the intersection graphs of rightward and downward rays. The collection of these rays is called a…

In this paper, we propose a new type of graph, denoted as "embedded-graph", and its theory, which employs a distributed representation to describe the relations on the graph edges. Embedded-graphs can express linguistic and complicated…

Let $k \geq 1$ be an integer. The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph~$G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. A…

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 $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$…

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let…

Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $\chi(F)=r+1$ and there is an edge $e$ in $F$ such that $\chi(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least…

A drawing of a graph $G$ is radial if the vertices of $G$ are placed on concentric circles $C_1, \ldots, C_k$ with common center $c$, and edges are drawn radially: every edge intersects every circle centered at $c$ at most once. $G$ is…

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for…

An arithmetical structure on a finite and connected graph G is a pair (d, r) of positive integer vectors such that r is primitive (the gcd of its entries is 1) and (diag(d) - A)r = 0, where A is the adjacency matrix of G. In this article,…

Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average…

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of $\G$ there are exactly $\gamma$ vertices of $\G$ which are…

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius $r$ is called an $r$-ASC graph. The $r$-ASC index $\theta_r(G)$ of a graph $G$ is the minimum number of vertices…

Given a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We prove a lower bound on the maximum spectral radius of unraveled balls of fixed radius,…