TU Wien:Diskrete Mathematik für Informatik UE (Gittenberger)/Übungen WS13/Beispiel 8

8) Prove the following statements: If G = (V, E) and G′= (V′, E′) are isomorphic graphs and φ :V→V′ is an isomorphism, then d_G(x) = d_G′(φ(x)) for all x ∈ V.

If, on the other hand, φ: V→V′ is a mapping satisfying d_G(x) = d_G′(φ(x)) for all x ∈ V, then G and G′ are not necessarily isomorphic.