We introduce the Chevalley cohomology for the graded Lie algebra of polyvector fields on $R^d$. This cohomology occurs naturally in the problem of construction and classification of fomalities on the sapce $ R^d$. Considering only graphs formalities, we define the Chevalley cohomology directly on spaces of graphs. We obtain some simple expressions for the Chevalley coboundary operator and we give examples and applications.