Neighborhood harmonic NH(G) = Σ 2/(S(u)+S(v)) over edges. S(v) = Σd(neighbors). Second-order version of classical harmonic H. Mondal-De-Pal (2019). NH emphasizes edges where BOTH endpoints have low neighborhood degree sums.
How to Use
Select graph:
- NH: Neighbor harmonic
- 2/(S+S): Per edge
- vs H: Compare
NH vs H
H = Σ2/(d+d): first-order. NH = Σ2/(S+S): second-order. NH is much smaller than H because S(v) ≥ d(v). NH discriminates better among graphs with similar degree sequences.
Bounds
For d-regular: NH = m·2/(2d²) = m/(d²). Compare H = m·2/(2d) = m/d. NH decreases as d² — more sensitive. NH(K_n) = n(n-1)/2 · 1/(n-1)² = n/(2(n-1)).
Step-by-Step Instructions
- 1Select graph.
- 2Compute S(v) for each vertex.
- 3For each edge: 2/(S(u)+S(v)).
- 4Sum all terms.
- 5Compare with H.