In this paper, we develop the theory of Sobolev spaces on locally finite
graphs, including completeness, reflexivity, separability, and Sobolev
inequalities. Since there is no exact concept of dimension on graphs, classical
methods that work on Euclidean spaces or Riemannian manifolds can not be
directly applied to graphs. To overcome this obstacle, we introduce a new
linear space composed of vector-valued functions with variable dimensions,
which is highly applicable for this issue on graphs and is uncommon when we
consider to apply the standard proofs on Euclidean spaces to Sobolev spaces on
graphs. 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. As fundamental analytical tools,
all these results would be extremely useful for partial differential equations
on locally finite graphs.Comment: 19 page