scholarly journals Monochromatic equilateral triangles in the unit distance graph

2020 ◽  
Vol 52 (4) ◽  
pp. 687-692
Author(s):  
Eric Naslund
Keyword(s):  
Distance Graph ◽  
Unit Distance ◽  
Unit Distance Graph ◽  
Equilateral Triangles

scholarly journals Distances in a rigid unit-distance graph in the plane

1991 ◽  
Vol 31 (2) ◽  
pp. 193-200 ◽  
Author(s):  
Hiroshi Maehara
Keyword(s):  
Distance Graph ◽  
Unit Distance ◽  
Unit Distance Graph

scholarly journals Distributed Dynamic Storage in Wireless Networks

2005 ◽  
Vol 1 (3-4) ◽  
pp. 355-371 ◽  
Author(s):  
Constantinos Georgiou ◽  
Evangelos Kranakis ◽  
Ricardo Marceí-n-Jiménez ◽  
Sergio Rajsbaum ◽  
Jorge Urrutia
Keyword(s):  
Spanning Trees ◽  
Distributed Storage ◽  
Network Size ◽  
Distance Graph ◽  
Unit Distance ◽  
Network Algorithms ◽  
Empty Intersection ◽  
Unit Distance Graph ◽  
Distributed Solutions ◽  
The Difference

This paper assumes a set of identical wireless hosts, each one aware of its location. The network is described by a unit distance graph whose vertices are points on the plane two of which are connected if their distance is at most one. The goal of this paper is to design local distributed solutions that require a constant number of communication rounds, independently of the network size or diameter. This is achieved through a combination of distributed computing and computational complexity tools. Starting with a unit distance graph, the paper shows: 1. How to extract a triangulated planar spanner; 2. Several algorithms are proposed to construct spanning trees of the triangulation. Also, it is described how to construct three spanning trees of the Delaunay triangulation having pairwise empty intersection, with high probability. These algorithms are interesting in their own right, since trees are a popular structure used by many network algorithms; 3. A load balanced distributed storage strategy on top of the trees is presented, that spreads replicas of data stored in the hosts in a way that the difference between the number of replicas stored by any two hosts is small. Each of the algorithms presented is local, and hence so is the final distributed storage solution, obtained by composing all of them. This implies that the solution adapts very quickly, in constant time, to network topology changes. We present a thorough experimental evaluation of each of the algorithms supporting our claims.

103.27 A property of a particular unit-distance graph

2019 ◽  
Vol 103 (557) ◽  
pp. 353-356
Author(s):  
Martin Griffiths
Keyword(s):  
Distance Graph ◽  
Unit Distance ◽  
Unit Distance Graph

On the dimension to represent a graph by a unit distance graph

1990 ◽  
Vol 6 (4) ◽  
pp. 365-367 ◽  
Author(s):  
Hiroshi Maehara ◽  
Vojtech Rödl
Keyword(s):  
Distance Graph ◽  
Unit Distance ◽  
Unit Distance Graph

scholarly journals Subdividing a Graph Toward a Unit-distance Graph in the Plane

2000 ◽  
Vol 21 (2) ◽  
pp. 223-229 ◽  
Author(s):  
Severino V. Gervacio ◽  
Hiroshi Maehara
Keyword(s):  
Distance Graph ◽  
Unit Distance ◽  
Unit Distance Graph

Periodic forests of stunted trees

1970 ◽  
Vol 266 (1172) ◽  
pp. 63-111 ◽  
Keyword(s):  
Infinite Plane ◽  
Base Period ◽  
Matrix Formulation ◽  
Unit Distance ◽  
Straight Line ◽  
Low Degree ◽  
Symmetry Properties ◽  
Equilateral Triangles ◽  
Basic Sets

We may define a forest of stunted trees as follows: Consider an infinite background of nodes at the vertices of an infinite plane tessellation of equilateral triangles., and start from a straight line of nodes at unit distance apart, which we shall consider as the ground ; other parallel lines of nodes are then spaced at successive levels of linearly increasing heights above the ground. Any node may be live (if a tree passes through it) or vacant otherwise. Any live node may give rise to a branch to one or other or both of the two nearest nodes at the next higher level, but this growth is stunted , on either side, if the neighbouring node on that side is also live and could provide a branch to the same higher level node (this other branch is also stunted). Many of the figures in the paper show the type of forest that results. The Introduction, § 1, describes the origin of this idea, and §2 gives definitions and points out certain basic properties and ideas for combining forests and for separating them into simpler units. A variety of periodicities is discussed. In §3 a mathematical theory is developed in terms of generating functions expressed as power series. Sequences and forests are represented by ratios ⌀(t)/f(t) of polynomials with coefficients in GF (2). A matrix formulation is also defined. The theory is developed in §4, so that periods and forests can be developed from those for basic sets having irreducible polynomials f(t) as denominators, with co-prime numerators of lower degree. In §5, the determination of base- and row-periods for particular irreducible polynomials f(t) is investigated as a preliminary to the enumeration of forests with given base-period n in §6, and of reflexive forests in §7. Further interesting properties, problems and applications are discussed in §8 ; it is intended to develop some of these in another paper. The tables give enumerations and properties connected with sequences and forests generated by various polynomials f(t) of low degree, culminating in table 5, which gives the numbers of forests with base periods up to 50, and table 6, which lists all individual forests with n up to 15. M any of these forests are given in the diagrams, intended to bring out various symmetry properties and possible variations.

scholarly journals Unit Distance Graphs and Algebraic Integers

2019 ◽  
Author(s):  
Danylo Radchenko
Keyword(s):  
Algebraic Integers ◽  
Unit Distance ◽  
Distance Graphs

scholarly journals Electric Vehicle Integration into Road Transportation, Intelligent Transportation, and Electric Power Systems: An Abu Dhabi Case Study

Smart Cities ◽  
2021 ◽  
Vol 4 (3) ◽  
pp. 1039-1057
Author(s):  
Amro M. Farid ◽  
Asha Viswanath ◽  
Reem Al-Junaibi ◽  
Deema Allan ◽  
Thomas J. T. Van der Van der Wardt
Keyword(s):  
Electric Power ◽  
Intelligent Transportation ◽  
Energy System ◽  
Transportation System ◽  
Electric Power System ◽  
Abu Dhabi ◽  
Road Transportation ◽  
Unit Distance ◽  
The Impact

Recently, electric vehicles (EV) have gained much attention as a potential enabling technology to support CO2 emissions reduction targets. Relative to their internal combustion vehicle counterparts, EVs consume less energy per unit distance, and add the benefit of not emitting any carbon dioxide in operation and instead shift their emissions to the existing local fleet of power generation. However, the true success of EVs depends on their successful integration with the supporting infrastructure systems. Building upon the recently published methodology for the same purpose, this paper presents a “systems-of-systems” case study assessing the impacts of EVs on these three systems in the context of Abu Dhabi. For the physical transportation system, a microscopic discrete-time traffic operations simulator is used to predict the kinematic state of the EV fleet over the duration of one day. For the impact on the intelligent transportation system (ITS), the integration of EVs into Abu Dhabi is studied using a multi-domain matrix (MDM) of the Abu Dhabi Department of Transportation ITS. Finally, for the impact on the electric power system, the EV traffic flow patterns from the CMS are used to calculate the timing and magnitude of charging loads. The paper concludes with the need for an intelligent transportation-energy system (ITES) which would coordinate traffic and energy management functionality.

New upper bound for the chromatic number of a random subgraph of a distance graph

2015 ◽  
Vol 97 (3-4) ◽  
pp. 326-332 ◽  
Author(s):  
A. S. Gusev
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Distance Graph ◽  
Random Subgraph
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