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 $G^{\tilde{w}}=(V,E,\tilde{w})$ is called a hyper-dual number weighted graph, where the weight $\tilde{w}(e)=w(e)+\mathrm{\Delta }w(e)(\epsilon +{\epsilon }^{\ast })$ is a hyper dual number, $\mathrm{\Delta }w(e)$ is a real number, $\epsilon$ and ${\epsilon }^{\ast }$ 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 $G^{\tilde{w}}$, 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.