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Extending the Tutte and Bollobás–Riordan polynomials to rank 3 weakly coloured stranded graphs

Part of: Graph theory Low-dimensional topology

Published online by Cambridge University Press:  25 October 2021

Remi C. Avohou ,
Joseph Ben Geloun  and
Mahouton N. Hounkonnou
Remi C. Avohou
Affiliation:
International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, University of Abomey-Calavi, 072BP50 Cotonou, Rep. of Benin Ecole Normale Supérieure de Natitingou, BP72 Natitingou, Rep. of Benin
Joseph Ben Geloun*
Affiliation:
International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, University of Abomey-Calavi, 072BP50 Cotonou, Rep. of Benin Université Paris 13, Sorbonne Paris Cité, 99, J.-B. Clément LIPN, Institut Galilée, CNRS UMR 7030, 93430, Villetaneuse, France
Mahouton N. Hounkonnou
Affiliation:
International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, University of Abomey-Calavi, 072BP50 Cotonou, Rep. of Benin
*
*Corresponding author. Email: joseph.bengeloun@gmail.com
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Abstract

The Bollobás–Riordan (BR) polynomial [(2002), Math. Ann.323 81] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly coloured stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs, which obeys a contraction/deletion recursion relation similar to that of the Tutte and BR polynamials. However, it is defined on a much broader class of objects, and furthermore captures properties that are not encoded by the Tutte or BR polynomials.

Keywords

Graph theoryTutte polynomialBollobás–Riordan polynomialtensor models

MSC classification

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Secondary: 57M15: Relations with graph theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 3 , May 2022 , pp. 507 - 549
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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