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

Yu Li, Lizhu Sun, Changjiang Bu

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 $\widetilde{w}(e)=w(e)+Δw(e)(\varepsilon+\varepsilon^{*})$ 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.

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

Abstract

Let be a positive-weighted graph with the weight for all . The weighted graph is called a hyper-dual number weighted graph, where the weight is a hyper dual number, is a real number, and are two dual units, . 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 , respectively. We establish quadratic forms of the Hessian matrices for the resistance distance and Kirchhoff index of 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 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.
Paper Structure (8 sections, 6 theorems, 73 equations)

This paper contains 8 sections, 6 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyper-dual numbers
  4. Hyper-dual number weighted graphs
  5. The Moore-Penrose inverse of hyper-dual number matrices
  6. Main result
  7. Resistance distance and Kirchhoff index of hyper-dual number weighted graphs
  8. Quadratic forms of Hessian matrices for resistance distance and Kirchhoff index of positive-weighted graphs and their estimations

Key Result

Theorem 2.2

Let $\widetilde{A}\in\mathbb{H}^{m\times n}$, $\widetilde{\mathbf{x}}\in\mathbb{H}^{n}$ and $\widetilde{\mathbf{b}}\in\mathbb{H}^{m}$. Suppose that $\widetilde{A}^{\dag}$ exists. Then the following statements hold. $(a)$ The solution to the hyper-dual equation $\widetilde{A}\widetilde{\mathbf{x}}=\w $(b)$ If the solution to the hyper-dual equation $\widetilde{A}\widetilde{\mathbf{x}}=\widetilde{\m

Theorems & Definitions (14)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • ...and 4 more