Generalization of the Bollobás-Riordan polynomial for tensor graphs

Abstract

Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial \documentclass[12pt]{minimal}\begin{document}$\mathcal T$\end{document}T encoding the supplementary topological information. This polynomial is a natural generalization of the Bollobás-Riordan polynomial (used to characterize matrix graphs) and is different from the Gurău polynomial [R. Gurău, Ann. Henri Poincare 11, 565 (2010)]10.1007/s00023-010-0035-6, defined for a particular class of tensor graphs, the colorable ones. The polynomial \documentclass[12pt]{minimal}\begin{document}$\mathcal T$\end{document}T is defined for both colorable and non-colorable graphs and it is proved to satisfy the deletion/contraction relation. A non-trivial example of a non-colorable graphs is analyzed.