We introduce a graph-theoretical interpretation of an induced action of Aut$(Γ)$ in the discrete de Rham cohomology of a finite graph $Γ$. This action produces a splitting of Aut$(Γ)$ that depends on the cycles of $Γ$. We also prove some graph-theoretical analogues of standard results in differential geometry, in particular, a graph version of Stokes' Theorem and the Mayer-Vietoris sequence in cohomology.