Quadratic form estimations for Hessian matrices of resistance distance and Kirchhoff index of positive-weighted graphs

Let $G^{w}=(V,E,w)$ be a positive-weighted graph with the weight $w(e)>0$ for all $e\in E$. The weighted graph $G^{\widetilde{w}}=(V,E,\widetilde{w})$ is called a hyper-dual number weighted graph, where the weight $\widetilde{w}(e)=w(e)+Δw(e)(\varepsilon+\varepsilon^{})$ is a hyper dual number, $Δw(e)$ is a real number, $\varepsilon$ and $\varepsilon^{}$ are two dual units, $e\in E$. In this paper, we give a representation for the Moore-Penrose inverse of the Laplacian matrix, and calculation formulas for the resistance distance and Kirchhoff index of $G^{\widetilde{w}}$, respectively. We establish quadratic forms of the Hessian matrices for the resistance distance and Kirchhoff index of $G^{w}$ via generalized matrix inverses. We further derive explicit bounds on the eigenvalues of the Hessian matrices for the resistance distance and the Kirchhoff index of $G^{w}$ in terms of graph parameters. We also prove that the Kirchhoff index of a positive-weighted graph with bounded edge weights is strongly convex on its edge weight vector.

💡 Research Summary

The paper investigates the second‑order sensitivity of two classical graph invariants—the resistance distance and the Kirchhoff index—on positively weighted graphs by exploiting hyper‑dual numbers. A hyper‑dual weighted graph G̃ is defined by assigning to each edge e a weight ŵ(e)=w(e)+Δw(e)(ε+ε*), where ε and ε* are dual units satisfying ε²=(ε*)²=εε*=εε=0. This representation enables automatic differentiation up to second order: the coefficient of εε in a hyper‑dual expression equals the quadratic form (Δx)ᵀ∇²f(x)Δx of the Hessian of the underlying real‑valued function f.

The authors first derive an explicit formula for the Moore‑Penrose inverse of the Laplacian of G̃. Let L be the Laplacian of the original weighted graph Gᵂ, and let L₁ be the real matrix induced by the perturbations Δw(e). Then the hyper‑dual Laplacian is L̃ = L + L₁(ε+ε*), and its Moore‑Penrose inverse is
 L̃† = L† − L†L₁L†(ε+ε*) + 2L†L₁L†L₁L† εε*.
Using this inverse, the resistance distance between vertices i and j in G̃ is expressed as
 R_{ij}(G̃) = (L†){ij} − (L†L₁L†){ij}(ε+ε*) + 2(L†L₁L†L₁L†)_{ij} εε*,
and the Kirchhoff index as
 K_f(G̃) = n·tr(L†) − n·tr(L†L₁L†)(ε+ε*) + 2n·tr(L†L₁L†L₁L†) εε*.

Extracting the εε* coefficients yields the exact quadratic forms of the Hessians for the original real‑valued functions R_{ij}(x) and K_f(x):
 (Δx)ᵀ∇²R_{ij}(x)Δx = 2·(L†L₁L†L₁L†)_{ij},
 (Δx)ᵀ∇²K_f(x)Δx = 2n·tr((L†)²L₁L†L₁).
Thus the Hessian matrices are expressed entirely in terms of the Moore‑Penrose inverse of the Laplacian and the perturbation matrix L₁.

The paper proceeds to bound the eigenvalues of these Hessians using classical graph parameters. Let λ₂ be the algebraic connectivity (second smallest Laplacian eigenvalue), λ_max the largest Laplacian eigenvalue, Δ_max the maximum weighted degree, and bR_{ij}=((L²)†)_{ij} the biharmonic distance. The authors prove that the smallest eigenvalue of ∇²K_f satisfies a lower bound proportional to λ₂⁴/(n·Δ_max²), while the largest eigenvalue is bounded above by a term involving λ_max³·Δ_max/(n·λ₂). Similar bounds are derived for the Hessian of each resistance distance, linking them to the biharmonic distances and the spectral gap. These results quantify how graph connectivity and degree heterogeneity control the curvature of the resistance‑based functionals.

A key theoretical contribution is the proof of strong convexity of the Kirchhoff index with respect to the edge‑weight vector when the weights lie in a bounded interval