Colin de Verdière invariant μ(G): spectral graph parameter with remarkable topological meaning. μ≤1 iff forest. μ≤2 iff outerplanar. μ≤3 iff planar (van der Holst-Lovász-Schrijver). μ≤4 iff linklessly embeddable. Monotone under minors.
How to Use
Select graph:
- μ: Invariant value
- Class: μ≤1,2,3,4
- Spectral: Matrix rank
Classification by μ
μ≤0: empty graph. μ≤1: forest (tree). μ≤2: outerplanar. μ≤3: planar. μ≤4: linklessly embeddable in R³. Each level corresponds to a fundamental topological graph class. Beautiful unifying framework.
Mathematical Theory
Introduced by Colin de Verdière (1990). Uses generalized Laplacian matrices with Strong Arnold Property. Minor-monotone: H minor of G → μ(H)≤μ(G). Connected to algebraic topology, spectral theory, and combinatorics.
Step-by-Step Instructions
- 1Select graph.
- 2Compute μ.
- 3Classify by μ.
- 4Check planarity.
- 5Apply minor monotonicity.