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. Moreover, we introduce a fractional Sobolev space, which is necessary when we study problems involving \((-\Delta )^{s}\) . Thirdly, we define the fractional divergence, and then give another form of \((-\Delta )^s\) , which leads to a formula of integration by parts. Finally, using the mountain-pass theorem and the Nehari manifold, we obtain multiplicity solutions to a discrete fractional Schrödinger equation. We caution the readers that though these existence results are well known in the continuous case, the discrete case is quite different.
