A note for W^{1,p}(V) and W_0^{1,p}(V) on a locally finite graph

In this paper, we investigate the Sobolev spaces W^{1,p}(V) and W_0^{1,p}(V) on a locally finite graph G=(V,E), which are fundamental tools when we apply the variational methods to partial differential equations on graphs. As a key contribution of this note, we show that in general, W^{1,p}(V) is not equivalent to W_0^{1,p}(V) on locally finite graphs, which is different from the situation on Euclidean space R^N.