Fractional Laplace operator on finite graphs

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.