The Distance Geometry Problem asks whether a given weighted graph has a realization in a target Euclidean space R which ensures that the Euclidean distance between two realized vertices incident to a same edge is equal to the given edge weight. In this paper we look at the setting where the target space is the surface of the sphere SK−1. We show that the DGP is almost the same in this setting, ...

Can an arbitrary graph be embedded in Euclidean space so that the isometry group of its vertex set is precisely its graph automorphism group? This paper gives an affirmative answer, explores the number of dimensions necessary, and classifies the outerplanar graphs that have such an embedding in the plane.

We consider the real three-dimensional Euclidean Jordan algebra associated to a strongly regular graph. Then, the Krein parameters of a strongly regular graph are generalized and some generalized Krein admissibility conditions are deduced. Furthermore, we establish some relations between the classical Krein parameters and the generalized Krein parameters.

We will present that the forbidden minors for 3-realizable graph are K5 and the octahedron. You can directly go to Theorem 13 and if necessary refer to the definitions, lemmas and theorems before it. Definition 1 A graph G is d-realizable if, given any realization p1, · · · , pn of the graph in some finite dimensional Euclidean space, there exists a realization q1, · · · , qn in E with the same...

The Harborth graph is the smallest known example of a 4-regular planar unit-distance graph. In this paper we give an analytical description of the coordinates of its vertices for a particular embedding in the Euclidean plane. More precisely, we show, how to calculate the minimal polynomials of the coordinates of its vertices (with the help of a computer algebra system), and list those. Furtherm...

A string p = p0p1 · · · pn−1 of non-negative integers is a Euclidean string if the string (p0 + 1)p1 · · · (pn−1 − 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + · · · + pn−1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Eucl...

The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R, and the goal is to choose k center points in R so that the sum of squared distances of each point to its nearest center is minimized. The best...

The geographical threshold graph model is a random graph model with nodes distributed in a Euclidean space and edges assigned through a function of distance and node weights. We study this model and give conditions for the absence and existence of the giant component, as well as for connectivity.

A unit disk graph is an intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers).

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes only cause a small damage remaining graph terms dilation, that is, property maintained for almost all residual graph. Constructions spanners near linear size are known low-dimensional Euclidean settings. Here, we present new constructions planar graphs, trees and (gen...