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

TL;DR

This paper develops quadratic form representations for the Hessian matrices of resistance distance and Kirchhoff index in weighted graphs, providing explicit eigenvalue bounds and demonstrating strong convexity of the Kirchhoff index.

Contribution

It introduces formulas for the Hessian matrices of resistance distance and Kirchhoff index using generalized inverses, and analyzes their eigenvalues and convexity properties.

Findings

Derived explicit formulas for the Hessian matrices.

Established eigenvalue bounds based on graph parameters.

Proved strong convexity of the Kirchhoff index under certain conditions.

Abstract

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 $\widetilde{w}(e)=w(e)+\Delta w(e)(\varepsilon+\varepsilon^{*})$ is a hyper dual number, $\Delta w(e)$ 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.