Generalization of the Bollobas-Riordan polynomial for tensor graphs

Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial T encoding the supplementary topological information. This polynomial is a natural generalization of the Bollobas-Riordan polynomial (used to characterize matrix graphs) and is different from the Gurau polynomial [R. Gurau, Ann. Henri Poincare 11, 565 (2010)], defined for a particular class of tensor graphs, the colorable ones. The polynomial 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. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3605312]