  We investigate recursive relations for the Grothendieck classes of the affine
graph hypersurface complements of melonic graphs. We compute these classes
explicitly for several families of melonic graphs, focusing on the case of
graphs with valence-$4$ internal vertices, relevant to CTKT tensor models. The
results hint at a complex and interesting structure, in terms of divisibility
relations or nontrivial relations between classes of graphs in different
families. Using the recursive relations we prove that the Grothendieck classes
of all melonic graphs are positive as polynomials in the class of the moduli
space $\mathcal M_{0,4}$. We also conjecture that the corresponding polynomials
are log-concave, on the basis of hundreds of explicit computations.
