Eigenvalue Analysis Examples

Graph Theory · Axiom Academy

EXAMPLE Eigenvalue Analysis Examples Computing eigenvalues, algebraic connectivity, Fiedler vectors, and spectral partitioning We'll find the eigenvalues of the complete graph on 4 vertices where every pair of vertices is connected. Excellent work! You've mastered spectral graph theory fundamentals. Here's what we learned: Adjacency Matrix Eigenvalues: For complete graph Kₙ, the eigenvalues are n-1 (once) and -1 (n-1 times). The largest eigenvalue relates to graph regularity and provides bounds on chromatic number. Laplacian Matrix: L = D - A where D is the degree matrix. The Laplacian is symmetric positive semi-definite with eigenvalues 0 = λ₁ ≤ λ₂ ≤ ... ≤ λₙ. Always has at least one zero eigenvalue. Algebraic Connectivity: The second-smallest Laplacian eigenvalue λ₂ (also called Fiedler value) measures how well-connected the graph is. If λ₂ = 0, the graph is disconnected. Larger λ₂ means better connectivity. Fiedler Vector: The eigenvector corresponding to λ₂ encodes structural information about the graph. Its components can be used to partition the graph into two well-separated communities. Spectral Partitioning: Sort vertices by Fiedler vector values and partition based on sign (or median). This gives an approximate solution to the NP-hard minimum bisection problem, widely used in clustering and community detection.

This is the written version of the interactive lesson above. See the full Graph Theory course.