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SOME RESULTS ON AN ALGEBRO-GEOMETRIC CONDITION ON GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  29 March 2017

AVI KULKARNI ,
GREGORY MAXEDON  and
KAREN YEATS
AVI KULKARNI
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC, Canada V5A 1S6 email akulkarn@sfu.ca
GREGORY MAXEDON
Affiliation:
Ballard Power Systems, 9000 Glenlyon Parkway, Burnaby BC, Canada V5J 5J8 email gregory.maxedon@gmail.com
KAREN YEATS*
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West, Waterloo ON, Canada N2L 3G1 email kayeats@uwaterloo.ca
*
✉kayeats@uwaterloo.ca
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Abstract

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Paolo Aluffi, inspired by an algebro-geometric problem, asked when the Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff polynomial of the same graph with one edge deleted. We give some results on which graph–edge pairs have this property. In particular, we show that multiple edges can be reduced to double edges, we characterize which edges of wheel graphs satisfy the property, we consider a stronger condition which guarantees the property for any parallel join, and we find a class of series–parallel graphs with the property.

Keywords

graphsKirchhoff polynomialJacobian ideal

MSC classification

Primary: 05C31: Graph polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 2 , April 2018 , pp. 218 - 254
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

All three authors were at Simon Fraser University at the time of this research. AK was supported by an NSERC PGS D. GM was supported by an NSERC USRA during this project. KY is supported by an NSERC discovery grant.

References

Aluffi, P., ‘Chern classes of graph hypersurfaces and deletion–contraction’, Mosc. Math. J. 12 (2012), 671–700.CrossRefGoogle Scholar
Aluffi, P. and Marcolli, M., ‘Algebro-geometric Feynman rules’, Int. J. Geom. Methods Mod. Phys. 8 (2011), 203–237.Google Scholar
Bloch, S., ‘Motives associated to graphs’, Jpn. J. Math. 2(1) (2007), 165–196.CrossRefGoogle Scholar
Bloch, S., Esnault, H. and Kreimer, D., ‘On motives associated to graph polynomials’, Commun. Math. Phys. 267 (2006), 181–225.CrossRefGoogle Scholar
Bogner, C. and Weinzierl, S., ‘Feynman graph polynomials’, Int. J. Mod. Phys. A 25 (2010), 2585–2618.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24(3–4) (1997), 235–265; Computational algebra and number theory (London, 1993).Google Scholar
Brown, F., ‘On the periods of some Feynman integrals’, Preprint, 2009, arXiv:0910.0114.Google Scholar
Brown, F. and Schnetz, O., ‘A K3 in 𝜙4 ’, Duke Math. J. 161(10) (2012), 1817–1862.CrossRefGoogle Scholar
Brown, F. and Schnetz, O., ‘Modular forms in quantum field theory’, Commun. Number Theory Phys. 7(2) (2013), 293–325.Google Scholar
Brown, F., Schnetz, O. and Yeats, K., ‘Properties of c 2 invariants of Feynman graphs’, Adv. Theor. Math. Phys. 18(2) (2014), 323–362.Google Scholar
Brown, F. and Yeats, K., ‘Spanning forest polynomials and the transcendental weight of Feynman graphs’, Commun. Math. Phys. 301(2) (2011), 357–382.Google Scholar
Doryn, D., ‘On one example and one counterexample in counting rational points on graph hypersurfaces’, Lett. Math. Phys. 97(3) (2011), 303–315.Google Scholar
Fulton, W., Algebraic Curves: An Introduction to Algebraic Geometry, Advanced Book Classics (Addison-Wesley, Advanced Book Program, Redwood City, CA, 1989).Google Scholar
Kulkarni, A., Maxedon, G. and Yeats, K., ‘Magma script accompanying this paper’, Available athttp://www.sfu.ca/∼akulkarn/Prop-616example.m, 2016, also available in the source of arXiv:1602.00356.Google Scholar
Marcolli, M., Feynman Motives (World Scientific, Singapore, 2010).Google Scholar
Panzer, E., ‘Hopf-algebraic renormalization of Kreimer’s toy model’, Master’s Thesis, Humboldt-Universität Zu Berlin, 2011.Google Scholar
Schnetz, O., ‘Quantum periods: A census of 𝜙4 -transcendentals’, Commun Number Theory Phys. 4(1) (2010), 1–48.Google Scholar