Related papers: Uniquely realisable graphs in analytic normed plan…
A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every…
For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs…
The problem of combinatorially determining the rank of the 3-dimensional bar-joint {\em rigidity matroid} of a graph is an important open problem in combinatorial rigidity theory. Maxwell's condition states that the edges of a graph $G=(V,…
We investigate a graph-theoretic problem motivated by questions in quantum computing concerning the propagation of information in quantum circuits. A graph $G$ is said to be a bounded extension of its subgraph $L$ if they share the same…
Jord\'an and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral…
Consider a network with $N$ nodes in $d$-dimensional Euclidean space, and $M$ subsets of these nodes $P_1,\cdots,P_M$. Assume that the nodes in a given $P_i$ are observed in a local coordinate system. The registration problem is to compute…
In this paper we establish combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks in the Euclidean plane for rotational groups of order 4 and 6, and of odd order between 5 and 1000, where a joint may lie at the…
We study a conductance-weighted arboricity for a finite simple undirected graph $G=(V,E,c)$ with a conductance assignment $c:E\to[0,\infty)$: \[ A_c(G):=\max\bigl\{ D_c(H): H\subseteq G\text{ connected}, |V(H)|\ge 2 \bigr\},\qquad…
Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary…
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 $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$…
The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary…
A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive.…
We study the problem of existence of two affine-equivalent bar-and-joint frameworks in Euclidean $d$-space which have prescribed combinatorial structure and edge lengths. We prove that theoretically this problem is always solvable, but we…
We prove a nonuniqueness theorem for Bernoulli site percolation on properly embedded planar graphs, and we obtain a general connectivity principle beyond planarity. Let $G$ be an infinite connected graph properly embedded in $\RR^2$ with…
Let G = (V, E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p1, p2, ..., pn} in R^d along with a one-to-one mapping…
We study the bar-and-joint frameworks in $\mathbb{R}^2$ such that some vertices are constrained to lie on some lines. The generic rigidity of such frameworks is characterised by Streinu and Theran (2010). Katoh and Tanigawa (2013) remarked…
Graph rigidity theory studies the capability of a graph embedded in the Euclidean space to constrain its global geometric shape via local constraints among nodes and edges, and has been widely exploited in network localization and formation…
This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss…
We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with…
The restricted edge-connectivity of a connected graph $G$, denoted by $\lambda^{\prime}(G)$, if it exists, is the minimum cardinality of a set of edges whose deletion makes $G$ disconnected and each component with at least 2 vertices. It…