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In Semi-supervised classification with Graph Convolutional Networks, I am unable to understand few things.

Given an undirected graph having adjacency matrix $A$, degree matrix $D_{ii} = \sum_j A_{ij}$
normalized graph laplacian $L = I_N + D^{-\frac{1}{2}}AD^{-\frac{1}{2}} = U \Lambda U^T$, where $\lambda_{max} \approx 2$ (see page 3, 2nd paragraph, not sure which matrix they are talking about)
Then, $I_N + D^{-\frac{1}{2}}AD^{-\frac{1}{2}}$ has eigenvalues in the range [0, 2]. How?

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The proof of the first statement can be found in these Lecture Notes.

Have a look of the Proof of the Claim 1.

Concerning the renormalization trick I do not see easy way to justify this statement. The paper claims:

i.e. adding self-loops to the graph, improves accuracy, and we demonstrate that this method effectively shrinks the graph spectral domain, resulting in a low-pass-type filter when applied to SGC.

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