Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractional Laplace operator on finite graphs

Mengjie Zhang, Yong Lin, Yunyan Yang

TL;DR

This work defines a discrete fractional Laplacian $(-\Delta)^s$ on finite graphs for all $s>0$ and shows it admits an explicit, computable representation in terms of Laplacian eigenpairs, with $(-\Delta)^s \phi_i=\lambda_i^s \phi_i$ and a matrix form. It distinguishes the regimes $0<s<1$ and $s>1$, giving a kernel-based form via $W_s$ for the former and a distributional, parity-dependent construction for the latter, while preserving a graph-appropriate integration by parts. The paper then analyzes the fractional Kazdan–Warner equation on finite graphs using variational methods and the method of upper and lower solutions, establishing solvability criteria dependent on the sign of the constant $c$ and the function $\kappa$, including a sharp threshold $c_{s,\kappa}$ and solvability at the threshold when finite. For $s>1$, analogous results are proved, extending the framework and maintaining the same qualitative solvability structure. Overall, the work provides a rigorously computable discrete analogue of the fractional Laplacian on finite graphs and applies it to nonlinear geometric equations on graphs with clear criteria and multiplicity results."

Abstract

Nowadays a great attention has been focused on the discrete fractional Laplace operator as the natural counterpart of the continuous one. In this paper, we discretize the fractional Laplace operator $(-Δ)^{s}$ for an arbitrary finite graph and any positive real number $s$. It is shown that $(-Δ)^{s}$ can be explicitly represented by eigenvalues and eigenfunctions of the Laplace operator $-Δ$. Moreover, we study its important properties, such as $(-Δ)^{s}$ converges to $-Δ$ as $s$ tends to $1$; while $(-Δ)^{s}$ converges to the identity map as $s$ tends to $0$ on a specific function space. For related problems involving the fractional Laplace operator, we consider the fractional Kazdan-Warner equation and obtain several existence results via variational principles and the method of upper and lower solutions.

Fractional Laplace operator on finite graphs

TL;DR

This work defines a discrete fractional Laplacian on finite graphs for all and shows it admits an explicit, computable representation in terms of Laplacian eigenpairs, with and a matrix form. It distinguishes the regimes and , giving a kernel-based form via for the former and a distributional, parity-dependent construction for the latter, while preserving a graph-appropriate integration by parts. The paper then analyzes the fractional Kazdan–Warner equation on finite graphs using variational methods and the method of upper and lower solutions, establishing solvability criteria dependent on the sign of the constant and the function , including a sharp threshold and solvability at the threshold when finite. For , analogous results are proved, extending the framework and maintaining the same qualitative solvability structure. Overall, the work provides a rigorously computable discrete analogue of the fractional Laplacian on finite graphs and applies it to nonlinear geometric equations on graphs with clear criteria and multiplicity results."

Abstract

Nowadays a great attention has been focused on the discrete fractional Laplace operator as the natural counterpart of the continuous one. In this paper, we discretize the fractional Laplace operator for an arbitrary finite graph and any positive real number . It is shown that can be explicitly represented by eigenvalues and eigenfunctions of the Laplace operator . Moreover, we study its important properties, such as converges to as tends to ; while converges to the identity map as tends to on a specific function space. For related problems involving the fractional Laplace operator, we consider the fractional Kazdan-Warner equation and obtain several existence results via variational principles and the method of upper and lower solutions.
Paper Structure (10 sections, 20 theorems, 157 equations)

This paper contains 10 sections, 20 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional Laplace operator
  4. The case $0<s<1$
  5. The case $s > 1$
  6. Fractional Kazdan-Warner equation
  7. Embeddings
  8. Proof of Theorem \ref{['T2']}
  9. Proof of Theorem \ref{['T4']}
  10. Proof of Theorem \ref{['T2-m']}

Key Result

Theorem 1.1

Let $0<s<1$, $c$ be a constant and $\kappa : V\rightarrow \mathbb{R}$ be a function. (i) If $c>0$, then (KW) is solvable if and only if $\kappa$ is positive somewhere. (ii) If $c=0$ and $\kappa \not\equiv 0$, then (KW) is solvable if and only if both $\kappa$ changes sign and $\int_{V} \kappa d \mu<

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 27 more