Why is the normalized Laplacian operator defined as $D^{-\frac{1}{2}}LD^{-\frac{1}{2}}$ (where $L = D - A$, with $D$ being the degree matrix of a graph and $A$ being the adjacency matrix of the graph) but not $D^{-1}L$? Are they not the same? Are there any reason to define it this way?
2026-03-27 14:03:26.1774620206
Why is the normalized Laplacian operator defined as $D^{-\frac{1}{2}}LD^{-\frac{1}{2}}$ but not $D^{-1}L$? Are they not the same?
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Because it’s symmetric, and so diagonalizable. This simplifies the linear algebra.
One could use the other matrix, but the symmetric version is a bit easier to work with. Note that since the two matrices are similar, they provide the same information.
If the graph is not regular, the matrices are not the same. (Try $P_3$.)