Counting of rational points on graph hypersurfaces

We will discuss special hypersufaces coming from Feynman integrals, their cohomology, class in the Grothendieck ring of varieties and the number of rational points over finite fields. I will mention the Kontsevich conjecture and a counterexample related to a K3 surface. I will explain the independence of the $c_2$ invariant from the representation space of the Feynman period.