Dot product dimension ρ(G): minimum d for mapping V→ℝ^d where u~v iff ⟨f(u),f(v)⟩ ≥ 1. Inner product threshold representation. Threshold graphs: ρ=1. Trees: ρ≤3. Every graph: ρ ≤ n-1.
How to Use
Select graph:
- ρ: Dot product dim
- ⟨·,·⟩: Inner product
- Threshold: ≥1
Representation
Each vertex v → vector f(v) in ℝ^d. u adjacent to v iff ⟨f(u), f(v)⟩ ≥ 1 (dot product at least 1). Captures angular proximity. Different from distance-based representations (boxicity, sphericity).
Bounds
Threshold graphs: ρ=1 (single dimension!). Trees: ρ≤3. Interval: ρ≤2. Planar: ρ = O(1) conjectured. General: ρ ≤ n-1. Computing ρ is NP-hard.
Step-by-Step Instructions
- 1Select graph.
- 2Compute ρ.
- 3Find embedding.
- 4Verify ⟨u,v⟩≥1.
- 5Compare dimensions.