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

As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the…

For a connected graph $G$, let $\mu(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by…

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a…

A. Reid showed that if $\Gamma_1$ and $\Gamma_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $\Gamma_1$ and $\Gamma_2$ are…

The maximum genus $\gamma_M(G)$ of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal…

We investigate the size of the embedded regular tree rooted at a vertex in a $d$ regular random graph. We show that almost always, the radius of this tree will be ${1/2}\log n$, where $n$ is the number of vertices in the graph. And we give…

A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining…

Let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic curves in $\mathbb{R}^d$. Assume that $\gamma_i$ is neither contained in a hyperplane nor in a quadric surface in $\mathbb{R}^d$, for each $i=1,2$. We show that for…

In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of…

Consider a positive integer $r$ and a graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges. The least $k$ so that a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ exists such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$…

Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H…

A $1$-factorization of a graph $G$ is a collection of edge-disjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$-factorization is that $|V(G)|$ is even and $G$ is regular; the converse is…

In a graph A, for each two arbitrary vertices g, h with d(g,h)=2,|MAg2h|=mAg2h is introduced the number of edges of A that are closer to g than to h. We say A is a 2-edge distance-balanced graph if we have mAg2h=mAh2g. In this article, we…

The \emph{genus} $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. A drawing of a graph on a surface is \emph{independently even} if every pair of nonadjacent…

The Euclidean dimension a graph $G$ is defined to be the smallest integer $d$ such that the vertices of $G$ can be located in $\mathbb{R}^d$ in such a way that two vertices are unit distance apart if and only if they are adjacent in $G$. In…

Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $\lambda_{1}(D(G))\geq\lambda_{2}(D(G))\geq\cdots\geq\lambda_{n}(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be…

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we…

The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if…