In recent years, considerable attention has been paid to the discrete fractional Laplace operator. In this paper, we define the fractional Laplace operator \((-\Delta )^{s}\) on arbitrary finite graphs for any \(s>0\) . Moreover, we investigate its properties, in particular its spectral characteristics for \(0<s<1\) and its limiting behavior as \(s\rightarrow 0^+\) and \(s\rightarrow 1^-\) . These properties differ significantly from the continuous setting. Finally, we address the solvability of a fractional Kazdan–Warner equation involving \((-\Delta )^{s}\) by means of variational methods and the method of upper and lower solutions.
