Heat Kernel & Spectral Geometry of Graphs

Visualize heat diffusion e^{−tL} on graphs — eigenmodes reveal geometry; "can you hear the shape?"

Graph: Size n: Time t: 0.50 Eigenmode: 1

Heat diffusion — node color = temperature

Selected Laplacian eigenmode

Eigenvalue spectrum λ₀≤λ₁≤⋯

Heat kernel trace tr(e^{−tL}) = Σ e^{−tλᵢ} vs time

The heat kernel H(t)=e^{−tL} governs diffusion: ∂u/∂t = −Lu. Kac (1966): "Can you hear the shape of a drum?" — the eigenvalue spectrum encodes geometry. For graphs: tr(H(t)) = Σᵢ e^{−tλᵢ} encodes node count, edge count, triangle count as t→0.