EN
Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach-Stone theorem is valid: if two Sobolev spaces on 3-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group 𝒢 and each p which is not an even integer, there exists n ∈ ℕ and a subspace $L ⊂ ℓⁿ_{p}$ whose group of isometries is the direct product 𝒢 × ℤ₂.