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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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References

Ambjorn, J., Durhuus, B. and Jonsson, T. (1991) Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6 1133.CrossRefGoogle Scholar
Avohou, R. C. (2016) Polynomial invariants for arbitrary rank D weakly-colored stranded graphs. SIGMA 12 030.Google Scholar
Avohou, R. C., Ben Geloun, J. and Livine, E. R. (2014) On terminal forms for topological polynomials for ribbon graphs: The N-petal flower. Eur. J. Comb. 36 348–366 [arXiv:1212.5961 [math.CO]].CrossRefGoogle Scholar
Avohou, R. C., Ben Geloun, J. and Nuwagira, B. Enumeration properties of the polynomial invariant for rank 3 weakly colored graphs (work in progress).Google Scholar
Avohou, R. C., Rivasseau, V. and Tanasa, A. (2015) Renormalization and Hopf algebraic structure of the five-dimensional quartic tensor field theory. J. Phys. A 48(48) 485204 [arXiv:1507.03548 [math-ph]].CrossRefGoogle Scholar
Ben Geloun, J., Magnen, J. and Rivasseau, V. (2010) Bosonic colored group field theory. Eur. Phys. J. C 70 1119 [arXiv:0911.1719 [hep-th]].CrossRefGoogle Scholar
Ben Geloun, J. and Rivasseau, V. (2013) A Renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 318 69 [arXiv:1111.4997 [hep-th]].CrossRefGoogle Scholar
Bollobás, B. (1998) Modern Graph Theory. Springer, NY.CrossRefGoogle Scholar
Bollobás, B. and Riordan, O. (2001) A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc. 83 513–531.Google Scholar
Bollobás, B. and Riordan, O. (2002) A polynomial of graphs on surfaces. Math. Ann. 323 81–96.CrossRefGoogle Scholar
Bonnington, C. P. and Little, C. H. C. (1995) The Foundations of Topological Graph Theory, 1st Ed. Springer-Verlag, New York.CrossRefGoogle Scholar
Bonzom, V., Gurau, R. and Rivasseau, V. (2012) Random tensor models in the large N limit: Uncoloring the colored tensor models. Phys. Rev. D 85 084037 [arXiv:1202.3637 [hep-th]].CrossRefGoogle Scholar
Caravelli, F. (2012) A simple proof of orientability in colored group field theory. SpringerPlus 1, 6 [arXiv:1012.4087 [math-ph]].CrossRefGoogle ScholarPubMed
Chmutov, S. (2009) Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial. J. Comb. Theory Ser. B 99 617–638. arXiv:0711.3490v3 [math.CO].CrossRefGoogle Scholar
Di Francesco, P., Ginsparg, P. H. and Zinn-Justin, J. (1995) 2-D Gravity and random matrices. Phys. Rept. 254 1 [arXiv:hep-th/9306153].CrossRefGoogle Scholar
Ellis-Monaghan, J. A. and Moffatt, I. (2013) Graphs on Surfaces Dualities, Polynomials, and Knots. Springer Briefs in Mathematics. Springer, NY.CrossRefGoogle Scholar
Gurau, R. (2011) Colored group field theory. Commun. Math. Phys. 304 69 [arXiv:0907.2582 [hep-th]].CrossRefGoogle Scholar
Gurau, R. (2010) Lost in translation: topological singularities in group field theory. Class. Quant. Grav. 27 235023 [arXiv:1006.0714 [hep-th]].CrossRefGoogle Scholar
Gurau, R. (2011) The 1/N expansion of colored tensor models. Annales Henri Poincare 12 829 [arXiv:1011.2726 [gr-qc]].CrossRefGoogle Scholar
Gurau, R. (2010) Topological graph polynomials in colored group field theory. Annales Henri Poincare 11 565 [arXiv:0911.1945 [hep-th]].CrossRefGoogle Scholar
Gurau, R. and Ryan, J. P. (2012) Colored Tensor Models - a review. SIGMA 8 020 [arXiv:1109.4812 [hep-th]].Google Scholar
Heffter, L. (1891) Über das problem der nachbargebiete. Math. Ann. 38 477–508.CrossRefGoogle Scholar
Krajewski, T., Rivasseau, V. and Vignes-Tourneret, F. (2011) Topological graph polynomials and quantum field theory. Part II. Mehler kernel theories. Annales Henri Poincare 12 483 [arXiv:0912.5438 [math-ph]].CrossRefGoogle Scholar
Krajewski, T., Rivasseau, V., Tanasa, A. and Wang, Z. (2011) Topological graph polynomials and quantum field theory. Part I. Heat kernel theories. Annales Henri Poincare ${\bf 12}$ 483 [arXiv:0811.0186v1 [math-ph]].CrossRefGoogle Scholar
Krushkal, V. and Renardy, D. A polynomial invariant and duality for triangulations. arXiv:1012.1310[math.CO].Google Scholar
Reshetikhin, N. Y. and Turaev, V. G. (1990) Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127 1–26.CrossRefGoogle Scholar
Raasakka, M. and Tanasa, A. (2014) Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory. Sem. Lothar. Comb. 70 B70d [arXiv:1306.1022 [gr-qc]].Google Scholar
Tanasa, A. (2011) Generalization of the Bollobás-Riordan polynomial for tensor graphs. J. Math. Phys. 52 073514 [arXiv:1012.1798 [math.CO]].CrossRefGoogle Scholar
Tutte, W. T. (1984) Graph Theory , Vol. 21 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Massachusetts.Google Scholar