Resolvent energy ER(G) = Σᵢ 1/(n-λᵢ). Uses resolvent function instead of exponential. ER ≥ 1 with equality iff G is edgeless. More sensitive to graph structure than ordinary energy for certain families. Applications in molecular branching.
How to Use
Select graph:
- ER: Resolvent energy
- 1/(n-λ): Per eigenvalue
- Bounds: Compare
ER vs Graph Energy
Graph energy: Σ|λᵢ|. Resolvent energy: Σ 1/(n-λᵢ). ER avoids absolute values → differentiates positive/negative eigenvalues. More discriminating for cospectral graphs. Better molecular descriptor in some cases.
Bounds
ER ≥ 1 (edgeless). ER(K_n) = 1/(n-n+1) + (n-1)/(n+1) = 1 + (n-1)/(n+1). For regular: ER = 1/(n-d) + (n-1)/(n+d/(n-1)). Clean formulas for many families.
Step-by-Step Instructions
- 1Select graph.
- 2Find eigenvalues.
- 3Sum 1/(n-λᵢ).
- 4Compare bounds.
- 5Interpret branching.