graph theory – monotone Fotler vector

Let $ u_0 geq u_1 geq cdots geq u_ {n-1} $ be positive numbers and define a matrix $ n times n $ by $ M_ {i, j} = u _ { left | i-j right |} $ for everyone $ i, j $.

Let $ L = D – M $ to be the Laplacian matrix of $ M $, or $ D $ is a diagonal matrix such that $ D_ {ii} = sum_j M_ {i, j} $ for everyone $ i $. Yes $ lambda_1 = 0 leq lambda_2 leq cdots leq lambda_n $ are the eigenvalues ​​of $ L $. Then we can show that $ lambda_2 $ has a monotone clean vector, called Fiedler vector.

(for proof see eg

If we put $ L = I – D ^ {- 1/2} MD ^ {- 1/2} $ is it possible to get the same result? It is that there is a monotone clean vector $ v $ corresponding to the second smallest eigenvalue of $ L & # 39;?

algorithms – Does the conversion of the representation of the adjacency matrix of a graph of size n x n into an adjacency list always require a time O (n ^ 2)?

"Nothing is always."

Without any prior knowledge, the worst complexity in time of converting an adjacency matrix representation of a graph into 0.1 values ​​into a corresponding adjacency list representation is $ Theta (n ^ 2) $.

However, if we know that the graph is a sparse matrix with a number of edges $ m = o (n ^ 2) $ and its adjacency matrix is ​​stored in a compressed format like the Yale format, so we can expect that the conversion will take $ o (n ^ 2) $ time.

Why does not the correct image appear with Facebook Open Graph?

I've set the Facebook Open Graph meta tags on my site, but when I use the Facebook debugger, the image I want to use is not displayed. I test it on the first image of the homepage of the mountain, but it is the image of the man with glasses instead)?

I included the og: image: secure_url information needed for an https: // connection.

I would like to add a specific image on my site that will soon be used by Facebook Open Graph, but I can not get an image of my choice?

Any help would be wonderful.

CommunityGraphPlot of the weighted graph: node too far

I play with weighted Graphic and CommunityGraphPlot, and I consider the following example.

listWeights = {2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1};
tab = {UndirectedEdge[3, 1], UndirectedEdge[3, 31], UndirectedEdge[3, 32], UndirectedEdge[3, 33]Table[UndirectedEdge[1, i + 1], {i, 1, 29}]} // Flatten
g = graph[tab[[All]], EdgeWeight -> Normalize @ listWeights]

which reproduce the following Graphic

enter the description of the image here

When I use the CommunityGraphPlot find communities

CommunityGraphPlot[g, ImageSize -> Full, VertexLabels -> "Name"]

I receive

enter the description of the image here

Question: why is the knot 2 traced until now? You can see in the code that it has an unstandardized weight of 2, so I expected it to be as close to the knot as possible 1.

Generalized Graph – Computer Stack Exchange

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Draw a graph with its vertices on a circle

If have the following code.

AdjacencyGraph[Table[Boole[CoprimeQ[i - j, 8]], {j, 0, 7}, {i, 0, 7}]]

This generates a graph with 8 vertices. If the vertices are labeled 0, 1, …, 7, where two vertices are adjacent each time their difference is relatively prime to 8 (so 3 and 6 are so since 6 – 3 = 3 is relatively prime to 8). I would like the graph to be drawn so that the vertices are in specific positions. Specifically, I would like the vertices 0, 1, …, 7 to be evenly distributed around a circle in that order. In other words, this graphic should look like a regular octagon with extra edges inside. Is there a way to modify my code so that vertices are drawn this way?

postgresql – Can PostGIS or Oracle Spatial and Graph be resized to hundreds of terabytes of data, or should I use a NoSQL option such as DynamoDB?

I'm going to have a very large dataset (maybe a few hundred terabytes), but it's very simple and very well structured data with very long points (that's why I want some GIS compatibility ).

From what I can say, PostGIS can only handle around 32 TB, and I do not know what Oracle Spatial can do for it. Amazon's DynamoDB can reach the petabyte, but I've read complaints that it can get pretty complicated and be avoided without having a good reason to use it. And using NoSQL for such structured data seems bad, but I find no other alternative for this size. On the cost side, yes, it's going to be expensive, but suppose it's not a big problem. The speed of recovery of spatial and temporal queries is the main decisive factor.

Questions about graph theory with connected graphs

a. Four vertices are labeled $ 1,2.3.4. How many ways can you draw edges between certain pairs of these vertices so that the result is a connected graph?

b. Five vertices are labeled $ 1,2,3,4.5. How many ways can you draw edges between certain pairs of these vertices so that the result is a connected graph?

I know this question has already been asked, but I did not understand the answer. I do not know how to solve this problem, so I would be grateful for help. – SDTT does not show that the first & # 39; hotel & # 39; in @graph

I have created Hotel schema, but want to include 3 as a list. The individual hotel scheme works but uses @graphic and trying to include a list does not do it.

I can not see what's wrong. The test tool only shows the first Hotel and not the second.

info visualization – A graph to display numerical data such as age

I want to show a variety of data points in a graph. For example, I have several data points, each representing the age of a user. What's a good graph representing "10 users are 18, 2 are 19, 1 is 20, 8 is 22 … etc."?

The problem is that the set of data points that I want to represent can vary considerably. For example, I might need to represent just 5 data points with extremely variable numbers (= ages) or several hundred data points, all between 20 and 30. Thus, the interval and the design of the graph must adapt to the data points that I have.

I would like to hear suggestions from people who have encountered the same problem.


I would like to use a graph that shows the absolute number of responses for each numeric value ("8 are 22 years old") instead of the percentage ("24% are 22 years old"). I think this can make the chart more difficult to understand but shows a lot more information because the number of answers can vary greatly and constitutes a huge part of the information I like to provide.