Fractional Laplace operator on finite graphs

TL;DR

This paper introduces a discretized version of the fractional Laplace operator on finite graphs, explores its properties, and applies it to solve related equations, bridging continuous and discrete fractional calculus.

Contribution

It provides an explicit eigenvalue-based representation of the fractional Laplace operator on finite graphs and analyzes its key properties and applications.

Findings

Fractional Laplace operator converges to the standard Laplacian as s approaches 1.

It converges to the identity operator as s approaches 0.

Existence results for fractional Kazdan-Warner equations are established.

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 $(-\Delta)^{s}$ for an arbitrary finite graph and any positive real number $s$. It is shown that $(-\Delta)^{s}$ can be explicitly represented by eigenvalues and eigenfunctions of the Laplace operator $-\Delta$. Moreover, we study its important properties, such as $(-\Delta)^{s}$ converges to $-\Delta$ as $s$ tends to $1$; while $(-\Delta)^{s}$ 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.