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How can the following labeled graph be extracted from the Laplacian Matrix below and viceversa?

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I had a look at this great conversation but it is already too advanced for me.

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They were very careful about labels. For off-diagonal matrix elements, let us call one $L_{ij}$ with $i \neq j,$ if vertex $i$ and vertex $j$ share an edge, then $L_{ij} = -1,$ if they do not share an edge then $L_{ij} = 0.$ The diagonal matrix element $L_{ii}$ is the valence of vertex $i,$ just the number of edges at that vertex.

The main properties of the matrix are: it is symmetric, the sum of all elements in a row is $0,$ and the sum of all elements in a column is $0.$ As a result, the vector with all entries $1$ is an eigenvector with eigenvalue $0.$ Other than that, I think the matrix comes out semidefinite, I can't quite remember.

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Thank you @Will Jagy do you know a website showing different examples of Laplacian Matrix with their labeled graphs? –  Arthur Mamou-Mani May 19 '12 at 19:55

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