How can the following labeled graph be extracted from the Laplacian Matrix below and viceversa?
I had a look at this great conversation but it is already too advanced for me.
2026-03-26 04:30:05.1774499405
Reading a Laplacian Matrix and its labeled graph?
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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.