All Questions
Tagged with pr.probability graph-theory
290 questions
1
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116
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How to use probability to find a matching in a family of graphs?
In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...
- 57
1
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0
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117
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Entropy of endpoints of a random walk in a dense graph
Let $p\in[0,1]$ be a constant and let $G$ be a graph with $n$ vertices and $\approx p\binom{n}{2}$ edges. If you'd like, consider $p=1/2$.
Let $X$ be a random vertex of $G$ chosen proportional to ...
- 761
3
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1
answer
108
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Expected size of matchings in a cubic graph
Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...
- 9,501
5
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1
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222
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Switching oriented paths in a graph
Consider an oriented graph (e.g. a finite part of the standard grid with some random orientations).
Each minute the following operation takes place: we choose uniformly randomly an ordered pair $(A,B)...
- 5,063
3
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1
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822
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Open Problems in Random Graphs [closed]
I am a PhD student in mathematics. I'm interested in probabilistic methods in combinatorics and especially random graphs. I am looking for an open problem in this area for my PhD proposal. I know that ...
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2
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1k
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Non-backtracking random walk in regular (finite) graphs
I know that many things are known when dealing with random walks on a finite (or even infinite) graph: mixing time, returns to origin, etc. All is based in the use of the Markovian property of such a ...
- 1,561
3
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1
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206
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Component properties in Euclidean graphs with distance threshold
In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given ...
- 649
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A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?
In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I ...
- 470
1
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0
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86
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A Random Graph Process
I'd like to understand the following random graph process. I'm not sure if it's difficult or straightforward, so apologies if this is below the level of mathoverflow, but I've gotten no response on ...
- 345
7
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171
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What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
- 1,462
2
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1
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607
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Component size distribution in small Erdos-Renyi networks
I'm looking at $\mathcal{G}(n,p)$ (I'll call these Erdos-Renyi networks) where $n$ is, say, at most 10.
I would like to know the probability a random node is in a component of size $m$.
It's ...
- 121
5
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1
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281
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Transfer-impedance matrix for edge correlations in random spanning tree
Suppose $G$ is a (weighted) connected graph and
let $T$ denote a random spanning tree of $G$,
chosen uniformly (or respecting the edge weights).
It is known that for any distinct edges $e, f$
$$\...
- 2,175
2
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1
answer
90
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Generalization: (The "number" of) smaller sized clusters in large random binary matrices follow a descending order. Why?
This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...
user125648
3
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1
answer
184
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Why is number of single cell clusters always greatest in a random matrix?
Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
user125648
1
vote
1
answer
114
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Probability for a group of stones to live on an infinite Go board
Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 ...
- 5,169
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83
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Growth models with lookahead
Given a connected graph $G$ with a connected subgraph $H$, we can consider the uniform distribution on the set of all sequences $H_0, H_1, \dots, H_r$ where $r = |E(G) \setminus E(H)|$, $H_0 = H$, $...
- 19.7k
2
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1
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508
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Proof and interpretation of the following percolation theory result for $n\times n$ square grid
While I was discussing this question with @JamesMartin, he mentioned a result here that:
In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such
that $\epsilon>0$ and $p_c$ is the ...
user123818
2
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1
answer
299
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Can this particular random matrix model be converted/related to any existing graph theory model?
Context:
This a sequel to the question: Is the Erdős–Rényi giant component result applicable here?
Consider a matrix whose elements are independently assigned a value
$1$ with probability $p$ ...
user123818
2
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2
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357
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Is the Erdős–Rényi giant component result applicable here?
Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$.
Define a cluster of cells as a maximal connected component in the ...
- 41
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173
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Why do larger random matrices maximize their number of clusters with lower densities?
Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$.
Define a cluster of cells as a maximal connected component in the ...
- 41
2
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103
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Is this correct: Inflection points of Euler number graph in Island-Mainland transition correspond to spanning cluster site percolation threshold?
I'm writing with respect to the paper Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice.
Here's a link to a PDF ...
user119567
4
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2
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4k
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Expected global clustering coefficient for Erdős–Rényi graph
What is the expected global clustering coefficient $\mathbb{E}[C_{GC}]$ for the Erdős–Rényi random graph (ER-graph) $\mathcal{G}(n,p)$ (expectation is over the ensemble of all ER-graphs) as $n \...
- 151
1
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188
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KPZ relation $\chi = 2 \xi -1$ in a random geometric graph
If I have $n$ points uniformly distributed on the surface of a torus, and form a graph by adding an edge between pairs whenever they are within a unit distance (induced by the Euclidean metric), I ...
- 640
2
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1
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232
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If the diameter of a bounded degree, directed graph is polynomial in the degree of the graph, is the mixing time also polynomial?
Given a directed graph $G=(V,E)$, with no self-loops, with a vertex that has a maximal out-degree $\le d\in O(\log |V|)$, and with a diameter $\text{diam}(G)\in O(\text{poly }d)$, consider converting ...
- 2,185
4
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1
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669
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Number of independent sets of a random tree
Let $T_n$ be a random tree on $n$ labelled vertices chosen equiprobably among all $n^{n - 2}$ trees, and $I(T)$ be the number of distinct independent sets of a tree $T$. I'm interested in the average ...
- 5,590
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2
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2k
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A random walk on an infinite graph is recurrent iff ...?
Q. Is there a master theorem that can be used to determine whether or not
a simple random walk (choose a random neighboring vertex as the next step)
on a given infinite graph
leads to ...
- 151k
0
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1
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181
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Bound on queries to a tree with unusual probabilties -- follow-up
This question follows up on Bound on queries to a tree with unusual probabilities, where @fedja was able to disprove my conjecture under only constraints (1-4) below. I restate the relevant facts here ...
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222
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Bound on queries to a tree with unusual probabilities
Consider a tree $\mathcal{T}(r) = (V,E)$ rooted at $r \in V$. Let $\kappa_r: V \longrightarrow [0,1]$ such that $\sum_{v \in V} \kappa_r(v)^2 = 1$. Furthermore, for any given vertex $v \neq r$, $\...
- 209
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343
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Cubic almost-vertex-transitive graphs with given spanning tree
Consider the infinite 3-regular tree. Pick a vertex $C$, the "center".
For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced ...
2
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0
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159
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Distribution of path probabilities for a finite absorbing Markov chain
I am interested in the distribution of path probabilities for a finite
absorbing (but otherwise well behaved) Markov chain. Has this topic
been considered in the literature?
A bit of Googling ...
- 15.4k
5
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2
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474
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Another graph characteristic
This question concerns a method of drawing graphs and a graph characteristic about which I want to learn more.
Consider a connected directed graph with at least one node with in-degree 0 and one node ...
- 9,720
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742
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Torus Graph Dynamics
Consider the torus graph, or the toroidal grid, which looks like
(The graph's vertices are the bold dots).
I will discuss only square torus graphs, where there is an equal number of vertices in a "...
- 403
4
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249
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Good introduction to Benjamini- Schramm limits [closed]
So I was wondering if someone might be able to suggest a good intro paper/ article for getting a feel for Benjamini- Schramm limits as well as getting a sense of the kinds of results that people have ...
- 662
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0
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98
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Asymptotic results on statistical graph models
This post is partly inspired by this post.
Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix
While it is well-known that two basic ...
- 8,071
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0
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1k
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The threshold for a perfect matching in a random subgraph of a regular bipartite graph?
The following question seems very natural.
It is a well known consequence of Hall's Theorem that every regular bipartite graph has a perfect matching. Another classical result states that the ...
- 1,643
4
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1
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317
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Infinite Tree with Poisson Clocks
Let $\mathcal{T}$ be the infinite countable $3$-regular tree graph. Pick a vertex in this graph, call it the root. Let the root carry the value $0$.
Next, assign $1$ to the neighbours of the root. ...
- 403
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0
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72
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A random variable standing for the size of connected component including a given node in a tree
Suppose we have a tree $T = (V,E)$, in which each nodes $v_i \in V$ has a probability $p_i$ to vanish. Let $v_0\in V$, we define random variable $\boldsymbol{X} = \boldsymbol{X}(T, v_0)$ stands for ...
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1
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0
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43
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a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
- 61
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463
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Expected number of connected components as $V(G)$ grows large
Let $E^c_n$ be the expected number of connected components of a simple undirected graph on the vertex set $\{1,\ldots,n\}$. (Every possible edge in $\big\{\{a, b\}: a, b\in \{1,\ldots,n\} \land a \neq ...
- 48.9k
7
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1
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222
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Algorithm to generate random commuting permutations
I am seeking to understand the properties of a typical pair of permutations $(\sigma,\tau) \in \mathrm{Sym}(n)^2$ chosen uniformly at random from all pairs such that $\sigma$ and $\tau$ commute. In ...
- 1,769
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1
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3k
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How to compute the clustering coefficient of a random graph?
How is the clustering coefficient defined for random graphs? For example, a first definition could be calling clustering coefficient of a random graph the expected value of the clustering coefficient ...
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2
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2k
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Graph with Poisson Clock at each Vertex
Let $G$ be a connected, undirected graph, with countably infinite set of vertices and countably infinite set of edges. Assume that the degree of each vertex is finite, and moreover, the degrees of all ...
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1
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1k
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Intuition on Kronecker Product of a Transition Matrix
Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider ...
- 1,421
3
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151
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Sequential generation of any random graph
The high-level question is: can we generate any random graph with size $d$ using a Markov chain?
For example, let $X^{(0)} = (1,0,\ldots,0) \in R^d$ be the initial state, and $X^{(t+1)} = f^{(t)}(X^{...
- 1,127
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141
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Fuzzy layers in graphs and neural networks
I wonder if the following statistical description of the layer architecture of finite graphs has been considered before and where I can find some references (especially under which name).
Consider a ...
- 9,720
5
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1
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479
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Expected Size of Independent Set
Q. Let $G = (V, E)$ be a graph with $V = \{v_1, \cdots, v_n\}$ and $E = \{(v_i, v_{i+1}) \mid 1 \leq i < n\}$. If we repeatedly remove vertices from $G$ uniformly randomly until the set of vertices ...
- 53
0
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0
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34
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What kind of prior on edge existence would form graphs that are unions of complete (sub)graphs?
Suppose a graph has $n$ vertices.
First question: is it possible to give a (nontrivial) prior probability on edge existence so that if a graph is created by querying the prior on the $\binom{n}{2}$ ...
- 118
1
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109
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Number of $H$-free graphs
Sorry if this is basic for MO. But the people at SE couldn't help me.
I'd like to get an estimate on the number of (labeled) $H$-free graphs on $n$ vertices, i.e. graphs in which no set of $|V(H)|$ ...
- 11
3
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1
answer
693
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Size of automorphism group of random regular graph
If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$?
--
I want to know what is the ...
- 13.9k
10
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0
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222
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Asymptotics of subgraph densities in graphons
In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...
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