Laplacian matrix
Properties

For an (undirected) graph G and its Laplacian matrix L with eigenvalues λ 0 ≤ λ 1 ≤ ⋯ ≤ λ n − 1 {\textstyle \lambda _{0}\leq \lambda _{1}\leq \cdots \leq \lambda _{n-1}}:

Because λ i {\textstyle \lambda _{i}} can be written as the inner product of the vector M v i {\textstyle M\mathbf {v} _{i}} with itself, this shows that λ i ≥ 0 {\textstyle \lambda _{i}\geq 0} and so the eigenvalues of L {\textstyle L} are all non-negative.

i.e., L rw {\textstyle L^{\text{rw}}} is similar to the normalized Laplacian L sym {\textstyle L^{\text{sym}}}. For this reason, even if L rw {\textstyle L^{\text{rw}}} is in general not symmetric, it has real eigenvalues — exactly the same as the eigenvalues of the normalized symmetric Laplacian L sym {\textstyle L^{\text{sym}}}.
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