Generalization of the Bollob\'as-Riordan polynomial for tensor graphs

TL;DR

This paper introduces a new polynomial invariant for tensor graphs, extending the Bollobás-Riordan polynomial to include non-colorable graphs, with potential applications in quantum gravity models.

Contribution

The paper generalizes the Bollobás-Riordan polynomial to all tensor graphs, including non-colorable ones, and proves it satisfies contraction/deletion relations.

Findings

Defines polynomial $\\mathcal T$ for tensor graphs

Extends polynomial to non-colorable graphs

Proves polynomial satisfies contraction/deletion relation

Abstract

Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial $\mathcal T$ encoding the supplementary topological information. This polynomial is a natural generalization of the Bollob\'as-Riordan polynomial (used to characterize matrix graphs) and is different of the Gur\uau polynomial, (R. Gur\uau, "Topological Graph Polynomials in Colored Group Field Theory", Annales Henri Poincare {\bf 11}, 565-584 (2010)) defined for a particular class of tensor graphs, the colorable ones. The polynomial $\mathcal T$ is defined for both colorable and non-colorable graphs and it is proved to satisfy the contraction/deletion relation. A non-trivial example of a non-colorable graphs is analyzed.