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Results tagged with random-graphs
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user 4558
The study of probability distributions over graphs. For example, the Erdős–Rényi model where each edge occurs independently with equal probability.
13
votes
2
answers
381
views
Comparing two measures on trees on $n$ vertices
A standard measure on trees on $n$ vertices is the Uniform Spanning Tree (UST) on the complete graph. This is the measure where every tree has equal probability, $1 / n^{n-2}$ by Cayley's formula.
H …
- 7,857
8
votes
1
answer
309
views
Longest induced cycles in random geometric graphs near criticality
We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge …
- 7,857
14
votes
5
answers
663
views
Do there exist sparse graphs with large crossing number?
Does there exist a sequence of graphs $\{ G_n \}$ such that
$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?
In particular, do rando …
- 7,857
57
votes
4
answers
15k
views
Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is asymptoti …
- 7,857