Motives of melonic graphs

  • Paolo Aluffi

    Florida State University, Tallahassee, USA
  • Matilde Marcolli

    California Institute of Technology, Pasadena, USA
  • Waleed Qaisar

    University of Toronto, Canada
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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- 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 . We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.

Cite this article

Paolo Aluffi, Matilde Marcolli, Waleed Qaisar, Motives of melonic graphs. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 10 (2023), no. 3, pp. 503–554

DOI 10.4171/AIHPD/156