Sobolev spaces on locally finite graphs

In this paper, we focus on the theory of Sobolev spaces on locally finite graphs, including completeness, reflexivity, separability and Sobolev inequalities. We introduce a linear space composed of vector-valued functions with variable dimensions such that the gradients of functions on graphs happen to fit into such a space and we can get the desired properties of various Sobolev spaces along this line. Moreover, we also derive several Sobolev inequalities under certain assumptions on measures or weights of graphs. Although these results are within the framework of functional analysis, the key is that we provide an appropriate perspective for applying variational methods on graphs. As fundamental analytical tools, all these results are highly applicable and useful for partial differential equations on locally finite graphs.