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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.

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