All Questions
Tagged with pr.probability graph-theory
290 questions
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Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
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51
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Subgraphs of random graphs with a given degree sequence
Let $\mathbf{d}=(d_1,\dots, d_n)$ be a given degree sequence with $3\leq d_i\leq \Delta$ for every $i$, where $\Delta$ is constant. Let $G(n,\mathbf{d})$ denote the random graph uniformly distributed ...
- 143
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2
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723
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Threshold function for a graph not being planar
A graph property $\mathcal{P}$ is monotone increasing if $G\in \mathcal{P}$ implies $G+e \in \mathcal{P}$, i.e., adding an edge to a graph does not destroy the property.
It is well-known that every ...
- 13.6k
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45
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Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
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72
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How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
3
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0
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81
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Can we remove the restriction on a parameter in Talagrand concentration inequality?
Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a ...
- 4,219
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5
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3k
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Simple random walk on a locally finite graph: when is it recurrent?
I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My ...
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63
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Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 12.9k
1
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0
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134
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Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
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2
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1
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199
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Average cluster size of a n-size vector
Given a vector of $n$ cells and $k$ elements in it, we can define a cluster of elements as a contiguous sequence of elements inside the vector.
My goal is to calculate the average cluster size for all ...
- 800
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1
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144
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Count arrangements with pairs of attacking kings [closed]
I have a $1\times n$ chessboard and $2$ pairs of kings in it. Both components of each pair of kings must be adjacent in the chessboard, that is, they must be attacking.
Now, I want to calculate the ...
- 13.4k
0
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55
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Counting matrix paths for (n,m>2) matrices
Given a $n\times m$ matrix with $k$ elements inside it, I need to calculate the number of arrangements of those $k$ elements that form at least 1 path from the top to bottom matrix row composed of the ...
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7
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2
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1k
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Why is the spectrum of Erdős–Renyi random graph approximately symmetric?
I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős–Renyi random graph $G(n,p)$ when $np\to\infty$,
and $np\to c=2,3$.
The plots above are already ...
- 5,405
3
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87
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Is the probability distribution of a graphon given as a graph limit computable?
Let $G_i$ be a sequence of finite graphs that is Cauchy in the space of graphons. That is, for every $\epsilon \in \mathbb Q_+$ there is a $N \in \mathbb N$ such that $$\forall n, m > N. \delta_\...
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84
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Percolative process distribution not equivalent to coupon collector problem distribution
I have a process where; given a $n\times 1$ matrix initially empty, an element is inserted in it at a random position, with the possibility of repeating the insertion at a filled cell. Then, after a ...
- 63.9k
18
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2
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1k
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In an Erdős–Rényi random graph, what is the threshold for the property "every edge is contained in at least one triangle"?
Let $G(n,p)$ denote the Erdős–Rényi random graph, where $n$ is the number of nodes and $p$ is the probability for each edge. I'm interested in precisely what range of $p$ the random graph has at least ...
CommunityBot
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100
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The uniform odd and even subgraph of $\mathbb{Z}^2$
Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
- 1,054
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1
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307
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How to estimate the fraction of graphs with small clique among the graphs with certain edges
Among all $n$-vertex graphs with $M$ edges and constant $k$, how to estimate the fraction of graphs of clique less than $k$? Thanks.
- 3,065
4
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1
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227
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Probability problem in Sheehan's conjecture
As my first math project, I have been working on Sheehan's Conjecture
and am stuck for weeks. I wonder if I am at a dead end.
Sheehan's Conjecture states that every Hamiltonian 4-regular simple
graph ...
- 3,499
4
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0
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1k
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Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
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4
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118
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Reorganizational matching
Motivation. My friend works in an organization that is re-organizing itself in the following somewhat laborious way: There are $n$ people currently sitting on $n$ jobs in total (everyone has one job). ...
- 48.9k
6
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1
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356
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Probabilistic problem on random spanning trees
Let $G(V,E)$ be a connected simple graph, where $V$ and $E$ denote respectively its vertex and the edge set respectively. Let $f: V\to \{-1,1\}$ a function mapping each vertex to a value in $\{-1,1\}$....
- 1,677
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2
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286
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Finding an easy example applying the general Lovász local lemma
Is there any easy application for the general local lemma as follows? If someone knows, please tell me the references or just post an example here. Thanks.
General Lovász local lemma: Consider a set $...
- 10.4k
2
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1
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156
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Some identities from graph theory and probability
The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...
- 656
4
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1
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594
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Martingales and intersection of random walks
Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
CommunityBot
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3
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2
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478
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Random spanning trees probability problem
We are given a simple connected graph $G(V,E)$ with vertex and edge set $V$ and $E$ respectively. For any vertex $v\in V$, let $D_T(v)$ the degree of $v$ in a uniformly generated random spanning tree $...
- 109k
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321
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Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?
In Bobkov and Tetali - Modified Log-Sobolev Inequalities, Mixing and Hypercontractivity (extended version Modified Logarithmic Sobolev Inequalities in Discrete Settings), at the beginning of section 3,...
- 12.9k
1
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62
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Nonintersecting witnesses of connectivity events in graphs
In my research I stumbled across a following result:
Let $G = (V, E)$ be a multigraph with three chosen vertices $a, b, c \in V$. We color its edges into red and blue colors: $E = R \sqcup S$. Events ...
7
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1
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382
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Diameter bound for graphs: spectral and random walk versions
This question can be phrased in different settings. I will discuss a spectral formulation and the equivalent random walk version. The question came up naturally in recent work with Devriendt and ...
- 14.2k
1
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0
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60
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Correct dependence for "Local Coloring"
In Alon-Spencer's book, Probabilistic Lens #8, it is proven that for each $k$, there exists $\epsilon = \epsilon(k)>0$ such that for all large $n$, there exists an $n$-vertex graph $G$ with ...
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2
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1
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115
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Randomly chosen walk of fixed length
Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v_1, \cdots, v_n)$ and $(w_1, \cdots, w_n)$ share an edge iff $\lvert v_i - w_i\rvert \leq 1$ for all $i$.
A walk of ...
- 17k
2
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1
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248
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Connected components in random regular graphs
Suppose we take a random regular graph $G_{2n, r}$, where $n$ is large. Let us also assume that $r$ is fixed, (not dependent on $n$). Let's say that half of the vertices of the graph are colored black ...
- 3,937
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0
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46
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Diameter of component graph of uniform spanning forests on the amenable transitive graph with super polynomial growth
According to the paper Benjamini, Kesten, Peres, and Schramm - Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12 (Annals, 2004), the diameter of the component graph of the ...
- 63.9k
1
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0
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86
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Min-sum belief propagation not working on a chain model with equal unary potentials
Given is a chain factor graph as presented in the image below with the following properties:
Each node can take values 0 or 1
All unary potentials are equal (e.g. $U(a)=0$) for every node $a$
All ...
51
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3
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4k
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What is the sandpile torsor?
Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
11
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2
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968
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Clique sizes in a unit disk graph
This is a spiritual successor to a question that Peter Shor answered here:
Generalized Euclidean TSP
Are there any results known on the asymptotic behavior of clique sizes in a unit disk graph with ...
- 1,446
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2
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2k
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Probability of Generating a Connected Graph
$N$ points are generated randomly within a unit square, with a uniform distribution.
What is the probability that the points form a connected graph, given that two points are connected if the distance ...
- 2,821
1
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1
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545
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Vertex degree on random graphs
Let $p = d/n$ with $d$ constant. How do I prove that, with high probability, $G_{n,p}$ contains a vertex of degree at least $(\log n)^{1/2}$,
where $G_{n,p}$ is a graph with $n$ vertices and the ...
CommunityBot
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1
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217
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Quasi-random vs pseudo-random graphs
My question is somehow concerning terminology on extremal graph theory.
Is there any difference concerning the notion of quasi-random graph and the notion of pseudo-random graph? My feeling is that ...
- 242
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190
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Probabilistic optimization problem on tree vertex selection without replacement proportional to the degree
We are given a tree $T(V,E)$ with $|V|=n$ vertices, where $V=\{v_1,v_2,\ldots, v_n\}$. We denote by $d_i$ the degree of vertex $v_i$ for all $i\in\{1,2,\ldots,n\}$.
In a sequential fashion, we select ...
2
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0
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215
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An approach to the prime number theorem with Rademacher variables and a recursive formula for the prime pi function?
Consider the bipartite graphs defined here:
Why is this bipartite graph a partial cube, if it is?
We do random walks on them with equal propability and since the graphs are finite and connected the ...
- 3,064
15
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4
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1k
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The critical value of percolation on Cayley graphs.
Let $\Gamma$ be a discrete group with a generating set $S$. Let $p_c(\Gamma,S)$ be the critical probability for percolation of the Cayley graph of $\Gamma$. Is it known that if $\Gamma$ is non-...
- 2,821
5
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1
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708
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First Passage Percolation on Trees
Let $T$ be a rooted Galton-Watson random tree generated accordingly to a probability distribution $\mu$. Now assign to each edge $e$ a random non-negative weight $w_e$ distributed a accordingly to a ...
- 2,821
6
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164
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Hamilton cycles in random graphs with just enough connectivity
What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...
- 7,545
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1k
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Optimal monotone families for the discrete isoperimetric inequality
Background: the discrete isoperimetric inequality
Start with a set $X=\{1,2,...,n\}$ of $n$ elements and the family $2^X$ of all subsets of $X$.
For a real number $p$ between zero and one, we consider ...
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15
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1
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1k
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Has the technique of "sprinkling" been used in studying random matrices?
In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...
- 114k
1
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1
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152
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Discrepancy of random bipartite graphs (2)
This question is a modification of the one asked here, which turned out to ask for something too strong to be true.
Given $k>0$ and a positive integer $n$, let $X, Y$ be two vertex sets of size $n$ ...
- 3,446
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1
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192
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Discrepancy of random bipartite graphs
This is a crosspost from MathStackExchange (original question).
Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$).
Define a ...
- 3,446
9
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1
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1k
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Vertex connectivity of random graphs?
Consider simple, undirected Erdős–Rényi graphs $G(n,p)$, where $n$ is the number of vertices and $p$ is the probability for each pair of vertices to form an edge. Many properties of these graphs are ...
- 1,354
1
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1
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183
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Expectation of edge weights on the complete graph
Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...
- 37.7k