All Questions
Tagged with pr.probability graph-theory
290 questions
1
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41
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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
2
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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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0
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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 ...
- 111
1
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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 ...
- 1,190
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0
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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. ...
- 47
1
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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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-3
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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 ...
- 47
2
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1
answer
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 ...
- 47
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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 ...
- 47
3
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0
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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_\...
- 6,353
1
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0
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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 ...
- 47
7
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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
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 ...
- 43
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
4
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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 ...
- 47
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, ...
- 1,305
2
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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 $...
- 1,190
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 ...
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 ...
- 3,499
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 ...
- 138
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 ...
- 1,407
6
votes
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
1
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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 ...
- 41
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 $...
- 1,677
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 ...
1
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1
answer
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 ...
- 21
4
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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 ...
- 1,561
3
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0
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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 ...
- 1,677
2
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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
6
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0
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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
6
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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 ...
- 557
1
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1
answer
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
3
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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
1
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1
answer
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 ...
- 26.9k
8
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2
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338
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Does entropy of the random walk control the return probability
Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n_x(y)$ is the probability ...
- 4,432
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 ...
- 715
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1
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128
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Non-linear diffusion on networks
The diffusion equation with constant diffusion $D$ can be represented as:
\begin{equation}
\frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t)
\end{equation}
where
$\Delta$ is the Laplace ...
- 117
5
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3
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840
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Probability of an edge in a random graph
Consider a vertex set $V$ and a degree sequence $(d_v)_{v\in V}$. I want to know the probability that an edge exists between two given vertices $u$ and $v$ in a random graph with this degree sequence.
...
- 1,444
2
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0
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90
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Odd $k$-cycle counts in graph with adjacency matrix $A$ is leading term in $\operatorname{tr} A^k$?
In a recent paper of Neeman, Radin, and Sadun, Moderate Deviations in Cycle Count, in the first line of section 7.3 they wrote $\tau_k(A)=\frac{\operatorname{tr}A^k}{n^k}+O(\frac 1n)$, but I don't ...
- 715
8
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0
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304
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"Meritocratic" pyramid schemes
There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...
- 2,075
2
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1
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165
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Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
- 153
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94
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"Cut norm" of conditional expectation has supremum on products of sets in sub-$\sigma$-algebra, or not?
I am reading Lovasz's book "Large networks and graph limits", and encountered the exercise that the stepping operator for graphons is contractive under the cut norm:
$$||W_P||_\square\leq||W|...
- 715
6
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1
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521
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Graphs resembling the math genealogy graph must have concentration in a small number of families?
I was talking with a non-mathematician the other week at a workshop about the fact that many mathematicians, like myself, are indexed in the math genealogy database. We talked a little about how many ...
- 3,209
3
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1
answer
161
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Probability permutation in turned to cycle
Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix).
If you permute the columns and rows independently what is the probability resulting permutation matrix is a ...
- 13.9k
1
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1
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119
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Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?
We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
1
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1
answer
173
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Could you provide some TSP examples from real world to test a new algorithm?
It's well known that to find a hamilton cycle is NPC, while TSP is NPH.
But it seems that for majority of graphs (density of edge > 0.1, order > 100) there is a fast algorithm to find different ...
- 175
1
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0
answers
340
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Random walk on non-abelian free group
Let $F_2$ be the free non-abelian group with generators $a, b\in F_2$.
Has the "random walk" where we start with the identity and then multiply it by $a$ or $b$ or $a^{-1}$ or $b^{-1}$ ...
- 11
2
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0
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344
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Probability that a graph and its complement are connected
It's well known that for any graph $G = (V,E)$ that if $G$ is not connected, then its compliment $\overline{G}$ is connected. So, it's impossible to have both $G$ and $\overline{G}$ be disconnected. ...
- 139