Kirchhoff matrix L = D - A: degree diagonal minus adjacency. Always positive semidefinite. Smallest eigenvalue = 0. Second smallest λ₂ = algebraic connectivity (Fiedler value). Number of zero eigenvalues = connected components.
How to Use
Select graph:
- L: Kirchhoff matrix
- λ: Eigenvalues
- τ: Cofactor = spanning trees
Spectral Properties
L is real symmetric → all eigenvalues real ≥ 0. λ₁=0 always. λ₂>0 iff connected (algebraic connectivity). λ_n ≤ 2Δ. For K_n: eigenvalues are 0 and n (multiplicity n-1). Fiedler vector partitions the graph.
Applications
Network reliability, random walks (hitting times), spectral clustering (Fiedler vector), graph partitioning, chip-firing games, electrical networks (effective resistance). The Laplacian is central to spectral graph theory.
Step-by-Step Instructions
- 1Select graph.
- 2Compute L=D-A.
- 3Find eigenvalues.
- 4Check connectivity.
- 5Get spanning trees.