Chapter 7: Symmetric Matrix. In the chapter mind map this icon labels Graph Laplacian: Spectral Methods & GNNs. The discussion below is excerpted and lightly edited from § Spectral Decomposition of Graph Laplacian in Mathematics for AI and Machine Learning.

Since $L$ is symmetric and positive semi-definite, it admits a spectral decomposition:

where $\Lambda = \mathrm{diag}(\lambda_0, \lambda_1, \ldots, \lambda_{N-1})$ with $0 = \lambda_0 \le \lambda_1 \le \cdots \le \lambda_{N-1}$, and $Q = [\mathbf q_0, \mathbf q_1, \ldots, \mathbf q_{N-1}]$ is orthonormal. The smallest eigenvalue $\lambda_0 = 0$ has eigenvector $\mathbf q_0 = \frac{1}{\sqrt{N}}\mathbf 1$ (constant vector) if the graph is connected. The number of zero eigenvalues equals the number of connected components in the graph.

The eigenvalues $\lambda_n$ are called the graph frequencies, and the eigenvectors $\mathbf q_n$ are the graph Fourier modes. This terminology comes from signal processing. The constant eigenvector $\mathbf q_0$ (corresponding to $\lambda_0 = 0$) represents the DC component (0 Hz), which is the average value of the signal across all nodes. The Fiedler vector $\mathbf q_1$ (corresponding to $\lambda_1$) is the lowest frequency wave, like a sine wave that crosses zero once, creating a natural bipartition of the graph. Large eigenvalues $\lambda_{N-1}$ correspond to high frequency noise, where adjacent nodes have alternating signs ($+$, $-$, $+$, $-$), representing rapid oscillations. Low frequencies correspond to smooth signals (slowly varying across the graph), while high frequencies correspond to oscillatory signals (rapidly varying).