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INFINITE RANDOM GEOMETRIC GRAPHS
"... Abstract. We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. If V is a countable dense set i ..."
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Cited by 5 (4 self)
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Abstract. We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. If V is a countable dense set
Hamiltonicity of the random geometric graph
, 2009
"... Let X1, X2,... be independent, uniformly random points from [0, 1] 2. For n ∈ N and r ≥ 0 the random geometric graph G(n, r) has vertex set Vn: = {X1,..., Xn} and an edge XiXj ∈ En iff. ‖Xi − Xj ‖ ≤ r. The ”hitting radius ” ρn(P) of an increasing graph property P is the least r such that G(n, r) sa ..."
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Cited by 1 (0 self)
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Let X1, X2,... be independent, uniformly random points from [0, 1] 2. For n ∈ N and r ≥ 0 the random geometric graph G(n, r) has vertex set Vn: = {X1,..., Xn} and an edge XiXj ∈ En iff. ‖Xi − Xj ‖ ≤ r. The ”hitting radius ” ρn(P) of an increasing graph property P is the least r such that G(n, r
Synchronization in Random Geometric Graphs
, 2008
"... In this paper we study the synchronization properties of random geometric graphs. We show that the onset of synchronization takes place roughly at the same value of the order parameter that a random graph with the same size and average connectivity. However, the dependence of the order parameter wit ..."
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Cited by 1 (1 self)
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In this paper we study the synchronization properties of random geometric graphs. We show that the onset of synchronization takes place roughly at the same value of the order parameter that a random graph with the same size and average connectivity. However, the dependence of the order parameter
Linear Orderings of Random Geometric Graphs
 GRAPH THEORETIC CONCEPTS IN COMPUTER SCIENCE
, 1997
"... In random geometric graphs, vertices are randomly distributed on [0, 1]² and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study sev ..."
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Cited by 7 (4 self)
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In random geometric graphs, vertices are randomly distributed on [0, 1]² and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study
Colouring random geometric graphs
"... some probability distribution ν on R d). For i � = j we join Xi and Xj by an edge if �Xi − Xj �< r(n). We study the properties of the chromatic number χn and clique number ωn of this graph as n becomes large, where we assume that r(n) → 0. We allow any choice ν that has a bounded density functio ..."
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some probability distribution ν on R d). For i � = j we join Xi and Xj by an edge if �Xi − Xj �< r(n). We study the properties of the chromatic number χn and clique number ωn of this graph as n becomes large, where we assume that r(n) → 0. We allow any choice ν that has a bounded density
The cover time of random geometric graphs
, 2009
"... We study the cover time of random geometric graphs. Let I(d) = [0,1] d denote the unit torus in d dimensions. Let D(x,r) denote the ball (disc) of radius r. Let Υd be the volume of the unit ball D(0,1) in d dimensions. A random geometric graph G = G(d,r,n) in d dimensions is defined as follows: Sam ..."
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Cited by 16 (3 self)
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We study the cover time of random geometric graphs. Let I(d) = [0,1] d denote the unit torus in d dimensions. Let D(x,r) denote the ball (disc) of radius r. Let Υd be the volume of the unit ball D(0,1) in d dimensions. A random geometric graph G = G(d,r,n) in d dimensions is defined as follows
On the Cover Time of Random Geometric Graphs
 In: ICALP. (2005
, 2005
"... Abstract. The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have be ..."
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Cited by 20 (3 self)
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Abstract. The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have
HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
, 2009
"... We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2connected. This proves a conjecture of Penrose. We also show that in the knearest neighbour model, there is a constant κ such that almost every κconnected graph ..."
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Cited by 4 (0 self)
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We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2connected. This proves a conjecture of Penrose. We also show that in the knearest neighbour model, there is a constant κ such that almost every κconnected graph
On the chromatic number of random geometric graphs
 Combinatorica
"... Given independent random points X1,..., Xn ∈ Rd with common probability distribution ν, and a positive distance r = r(n)> 0, we construct a random geometric graph Gn with vertex set {1,..., n} where distinct i and j are adjacent when ‖Xi − Xj ‖ ≤ r. Here ‖. ‖ may be any norm on Rd, and ν may be ..."
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Cited by 13 (4 self)
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Given independent random points X1,..., Xn ∈ Rd with common probability distribution ν, and a positive distance r = r(n)> 0, we construct a random geometric graph Gn with vertex set {1,..., n} where distinct i and j are adjacent when ‖Xi − Xj ‖ ≤ r. Here ‖. ‖ may be any norm on Rd, and ν may
Clique number of random geometric graphs
, 2013
"... The clique number C of a graph is the largest clique size in the graph. For a random geometric graph of n vertices, taken uniformly at random, including an edge beween two vertices if their distance, taken with the uniform norm, is less than a parameter r on a torus T d a, we find the asymptotic beh ..."
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The clique number C of a graph is the largest clique size in the graph. For a random geometric graph of n vertices, taken uniformly at random, including an edge beween two vertices if their distance, taken with the uniform norm, is less than a parameter r on a torus T d a, we find the asymptotic
Results 1  10
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976,448