The Königsberg Bridges Meet Graph Laplacians

From Euler’s bridges, we now have spectral graph theory. The Laplacian matrix L = D – A (diagonal of degrees minus adjacency). Its second smallest eigenvalue λ₂ (the algebraic connectivity) measures how well‑connected a graph is. For a path graph with 10 vertices, λ₂ is small; for a complete graph, λ₂ = n. λ₂ is used in image segmentation (spectral clustering), in designing robust networks, and in understanding synchronisation of oscillators (e.g., fireflies or power grids). Euler’s problem grew into a field that analyses Facebook friends and brain networks.