When we represent a network of sensors in Euclidean space by a graph, there are two distances between any two nodes that we may consider. One of them is the Euclidean distance. The other is the distance between the two nodes in the graph, defined to be the number of edges on a shortest path between them. In this paper, we consider a network of sensors placed uniformly at random in a two-dimensi...

A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclide...

Rigidity is the property of a structure that does not flex. It well studied in discrete geometry and mechanics, has applications material science, engineering biological sciences. A bar-and-joint framework pair (G,p) graph G together with map p vertices into Euclidean space. We view edges as bars universal joints. The can move continuously long distances between pairs adjacent are preserved. ri...

Chang, MS., C.Y. Tang and R.C.T. Lee, Solving the Euclidean bottleneck biconnected edge subgraph problem by 2-relative neighborhood graphs, Discrete Applied Mathematics 39 (1992) 1-12. In this paper, we discuss the Euclidean bottleneck biconnected edge subgraph problem. We shall first define a k-relative neighborhood graph which is similar to the relative neighborhood graph first proposed by To...

In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space (R with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rig...

In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of I-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each I-graph I(n, j, k) admits a unit-distance representation in the Euclidean plane. This implies tha...