Fractional Laplace operator and related Schr\"odinger equations on locally finite graphs

In this paper, we first define a discrete version of the fractional Laplace operator $(-\Delta)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, \mu, w)$. Secondly, we define the fractional divergence and give another form of $(-\Delta)^s$. The third point, and the foremost, is the introduction of the fractional Sobolev space $W^{s,2}(V)$, which is necessary when we study problems involving $(-\Delta)^{s}$. Finally, using the mountain-pass theorem and the Nehari manifold, we obtain multiplicity solutions to a discrete fractional Schr\"{o}dinger equation on $G$. We caution the readers that though these existence results are well known in the continuous case, the discrete case is quite different.