### Citations

2990 | On random graphs
- Erdős, Rényi
- 1959
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Citation Context ...on. In this presentation I will give a survey over some recent results on random graphs where I have been at least partly involved. The systematic study of random graphs was started by Erdos and Rnyi =-=[4]-=- in 1960, and the theory has expanded rapidly during the last decade. For a fuller historical account, and for many other results on random graphs, I refer to Bollobs's book [3]. Definitions A random ... |

39 |
A central limit theorem for decomposable random variables with applications to random graphs
- Barbour, Karoński, et al.
- 1989
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Citation Context .... (An induced subgraph is obtained by selecting a subset of the vertices together with all edges between them in the graph; a general subgraph on the same vertices may contain fewer edges.) Theorem 4 =-=[2,11,13]-=-. For the number X n of induced subgraphs isomorphic to H in G n;p , p 2 (0; 1) fixed, the following holds. (i) If p 6= p H = e= \Gamma n 2 \Delta , then Var X n i n 2v\Gamma2 and (1) holds. (ii) If p... |

36 |
The diameter of random graphs
- BOLLOBS
- 1981
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Citation Context ...rted by Erdos and Rnyi [4] in 1960, and the theory has expanded rapidly during the last decade. For a fuller historical account, and for many other results on random graphs, I refer to Bollobs's book =-=[3]-=-. Definitions A random graph is a graph generated by some random procedure. There are many (non-equivalent) ways to define random graphs. The simplest, denoted by G n;m (or one of several common simil... |

34 |
The first cycles in an evolving graph
- Flajolet, Knuth, et al.
- 1989
(Show Context)
Citation Context .... Then P(L n = k) ! p k = 1 2 Z 1 0 t k\Gamma1 (1 \Gamma t) 1=2 e t=2+t 2 =4 dt; ks3: The limit distribution has infinite mean, in fact p k i k \Gamma3=2 , and thus EL n !1. More precisely: Theorem 2 =-=[5,15]. EL n �-=-��� Kn 1=6 , where K = 2 \Gamma1=6 3 \Gamma2=3 e 3=4 �� 1=2 \Gamma(1=3). Secondly, let us study the phase transition more closely. It turns out that the correct scale for doing this is to let m ... |

29 |
When are small subgraphs of a random graph normally distributed? Probab. Theory Related Fields 78
- Ruciński
- 1988
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Citation Context ..., however; for the variable defined above, some restriction on the structure of H is needed for the Poisson result and for the normal result when p is small, and we need 1 \Gamma p AE n \Gamma2 ; see =-=[3,18,17]-=-. If p is fixed, there are no problems. Let v and es1 be the number of vertices and edges in H. Then Var X n i n 2v\Gamma2 , and X n \Gamma EX n (Var X n ) 1=2 d \Gamma! N(0; 1): (1) 4 SVANTE JANSON I... |

27 | The number of spanning trees, Hamilton cycles, and perfect matchings in a random graph - Janson - 1994 |

25 |
Random coverings in several dimensions
- Janson
- 1986
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Citation Context ...unction spaces, see for example [6,7], but gradually I became more and more interested in probability theory. Some years ago I worked on random coverings and other problems in geometrical probability =-=[8,9]-=-, and at present most of my time is devoted to the related field of combinatorial probability, in particular random graphs. This may seem to be very different from the harmonic and functional analysis... |

19 | On functions with conditions on the mean oscillation - Janson - 1976 |

19 |
The birth of the giant component. Random Struct
- Janson, Knuth, et al.
- 1993
(Show Context)
Citation Context .... Then P(L n = k) ! p k = 1 2 Z 1 0 t k\Gamma1 (1 \Gamma t) 1=2 e t=2+t 2 =4 dt; ks3: The limit distribution has infinite mean, in fact p k i k \Gamma3=2 , and thus EL n !1. More precisely: Theorem 2 =-=[5,15]. EL n �-=-��� Kn 1=6 , where K = 2 \Gamma1=6 3 \Gamma2=3 e 3=4 �� 1=2 \Gamma(1=3). Secondly, let us study the phase transition more closely. It turns out that the correct scale for doing this is to let m ... |

16 |
Poisson convergence and Poisson processes with applications to random graphs. Stochastic Process
- Janson
- 1987
(Show Context)
Citation Context ...k closer at two phases of this evolution. First, what is the size of the first cycle that appears? It turns out that the length converges in distribution as n !1, without any normalization. Theorem 1 =-=[10]-=-. Let L n be the length of the first cycle. Then P(L n = k) ! p k = 1 2 Z 1 0 t k\Gamma1 (1 \Gamma t) 1=2 e t=2+t 2 =4 dt; ks3: The limit distribution has infinite mean, in fact p k i k \Gamma3=2 , an... |

14 |
Multicyclic components in random graphs process. Random Structures and Algorithms
- Janson
- 1993
(Show Context)
Citation Context ... is always exactly one, the giant component. During this range, however, it is possible that there are several components with more than one cycle, and we have the following precise result. Theorem 3 =-=[12,15]. The pr-=-obability that an evolving random graph never has more than one multicyclic component during the entire evolution approaches 5�� 18 �� 0:8727 as n !1. Subgraph counts Another class of results ... |

14 |
Orthogonal decompositions and functional limit theorems for random graph statistics
- Janson
- 1994
(Show Context)
Citation Context ...ves a random graph G n;p at a fixed time p 2 (0; 1) and a graph G n;m at the random time at which the m:th edge appears. Results for both models may then be obtained from results for the process, see =-=[13]-=-. It is also of interest to study properties of the whole process, such as the first time something happens or the maximum of some variable during the evolution. The structure and evolution of the com... |

10 |
A functional limit theorem for random graphs with applications to subgraph count statistics, Random Struct. Algorithms 1
- Janson
- 1990
(Show Context)
Citation Context .... (An induced subgraph is obtained by selecting a subset of the vertices together with all edges between them in the graph; a general subgraph on the same vertices may contain fewer edges.) Theorem 4 =-=[2,11,13]-=-. For the number X n of induced subgraphs isomorphic to H in G n;p , p 2 (0; 1) fixed, the following holds. (i) If p 6= p H = e= \Gamma n 2 \Delta , then Var X n i n 2v\Gamma2 and (1) holds. (ii) If p... |

2 |
Balanced graphs and the problem of subgraphs of random graphs
- Ruciński, Vince
- 1985
(Show Context)
Citation Context ..., however; for the variable defined above, some restriction on the structure of H is needed for the Poisson result and for the normal result when p is small, and we need 1 \Gamma p AE n \Gamma2 ; see =-=[3,18,17]-=-. If p is fixed, there are no problems. Let v and es1 be the number of vertices and edges in H. Then Var X n i n 2v\Gamma2 , and X n \Gamma EX n (Var X n ) 1=2 d \Gamma! N(0; 1): (1) 4 SVANTE JANSON I... |

1 | On BMO and related spaces - Janson - 1977 |

1 |
Random coverings and related problems
- Janson
- 1984
(Show Context)
Citation Context ...unction spaces, see for example [6,7], but gradually I became more and more interested in probability theory. Some years ago I worked on random coverings and other problems in geometrical probability =-=[8,9]-=-, and at present most of my time is devoted to the related field of combinatorial probability, in particular random graphs. This may seem to be very different from the harmonic and functional analysis... |

1 |
An example of a superproportional graph
- Krrman
- 1991
(Show Context)
Citation Context ...nt and c 2 ? 0. The exceptional graphs that give case (iii) have a combinatorial description and are called proportional. The smallest examples have 8 vertices, for example the wheel W 8 . Jan Krrman =-=[16]-=- has found an example with 64 vertices where furthermore c 1 = 0. For induced subgraph counts in G n;m , with m = [ \Gamma n 2 \Delta p] for a fixed p, the situation is similar but not identical. Ther... |