## 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 https://arxiv.org/abs/1411.0210).

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 inset.agency 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

When I use the CommunityGraphPlot find communities

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


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.

## schema.org – 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.


{
"@context":"http://schema.org",
"@graph":[
{
"@type":"Hotel",
"name" : "The Hampshire Hotel",
"description" : "A boutique hotel with 42 rooms, excellent value for money located just outside of St Helier.",
"addressLocality" : "St Helier",
"postalCode" : "JE2 4TB",
"streetAddress" : "53 Val Plaisant"
},
"photo" : "TBC our image link",
"telephone" :  "+441534 724 115",
"PetsAllowed" : "No",
"priceRange" : "£49 - £100",
"starRating" :  "3*",
"amenityFeature" : [
"Swimming Pool",
"Bar & restaurant",
"Car parking",
"24hour service",
"Free Wifi"
]
}
]
},
{
"@type":"Hotel",
"name" : "The Club Hotel & Spa",
"description" : "A luxury boutique hotel in the heart of St Helier, offering outstanding restaurant, spa and swimming pools.",
"addressLocality" : "St Helier",
"postalCode" : "JE2 4UH",
"streetAddress" : "Green Street"
},
"photo" : "TBC our image link",
"telephone" :  "+44 1534 876500",
"starRating" :  "5*",
"PetsAllowed" : "No",
"priceRange" : "£83 - £224",
"amenityFeature" : [
"Swimming Pools",
"Spa",
"Bar & restaurant",
"24hour service",
"Free Wifi"
]
}
]
}



## 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.

Update

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.