Graph polynomials and their applications

Speaker: Joanna A. Ellis-Monaghan
Topics: Graph polynomials and their applications
Tentative Schedule
Week One: Deletion-contraction models. The Tutte polynomial is perhaps the most
studied and important graph polynomial, and so a natural starting point. We will use the
Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in
general. These include several ways in which a graph polynomial may be defined and methods
for extracting combinatorial information and algebraic properties from it. En route, we
will encounter several other graph polynomials, such as the chromatic, flow, reliability, and
shelling polynomials, which can be shown to be evaluations of the Tutte polynomial. We
also use the Tutte polynomial to demonstrate how graph polynomials may be generalized,
by expanding either the domain or the parameter space. The Tutte polynomial has applications
in many fields, which we will see here with a particularly fruitful connection with the
Potts model of physics. We conclude with a brief discussion of computational complexity
considerations.
Day 1 Intro to graph polynomials. Creating them and extracting combinatorial information.
Day 2 The Tutte polynomial. Several definitions and universality.
Day 3 Evaluations and specializations of the Tutte polynomial. Chromatic, flow, reliability,
derivatives, and graph statistics.
Day 4 Generalizations of the Tutte polynomial. Multivariate and topological extensions.
Day 5 Applications of the Tutte polynomial: The Potts model and computational complexity.
Week Two: Skein Models. As ubiquitous as the Tutte polynomial is, there are nevertheless
many other graph polynomials which are not evaluations of it. We turn now to
an important class of graph polynomials, those that maybe computed via, not deletioncontraction,
but rather skein-type reductions, similar to those of knot theory. For this class
of graph polynomials, the generalized transition polynomial plays the universal role that the
Tutte polynomial does for deletion-contraction invariants, and we see that the Penrose and
circuit partition polynomials are specializations of it. We will also see that the transition
polynomial is particularly well-adapted to encoding topological information. While these
polynomials are not generally specializations of the Tutte polynomial, there are often surprising
connections among them. For example, the interlace polynomial, computed neither
by deletion-contraction nor a skein relation, but rather a pivot operation, is related to the
Tutte polynomial through the transition polynomial of a medial graph. We emphasize these
interrelations which enable results for one polynomial to give results for others. Again, there
are many applications of these polynomials, and we conclude with three examples, two in
the area of DNA structures, and one with the interplay with knot theory.
Day 1 Skein-type polynomials. The Transition, Martin, and Penrose polynomials.
Day 2 The generalized transition polynomial. Multivariate and topological extensions.
Day 3 Interconnections. The interlace polynomial and interrelations among different polynomials.
Day 4 Applications to DNA sequencing and DNA self-assembly:
Day 5 Connections with knot theory. Recovering the Kauffmann bracket.
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A sample range of questions related to this topic
Big questions. These are known to be more or less NP-hard in the most general case,
but there are many, many special variations to consider. People have made careers out of
investigating these area.
• Find exact solutions to the Tutte/Potts model for various lattices and various models.
• Determine conditions for sequence reconstruction with various levels of data.
Focused open questions. These questions are easy to state, but they have been open for
a while.
• Show that the U (or W) polynomial is a complete invariant of trees (or even at least
caterpillars).
• Prove that P(G;-1) is non-zero for plane graphs.
• Do the coefficients of the Penrose polynomial alternate in sign?
Straightforward exercises. These are known results, more or less on the level found in a
typical graduate text.
• Find formulas for the chromatic polynomial for common classes of graphs (paths,
cycles, complete graphs, etc.)
• Prove that the reliability polynomial is a specialization of the Tutte polynomial
• Compute the Potts model for a small grid, and use it to find the probability of a
monochromatic state.
A partial list of resources for the course
[1] L. Beaudin*, J. Ellis-Monaghan, G. Pangborn, R. Shrock, A little statistical mechanics for the graph
theorist, Discrete Mathematics, 310 (13-14) 2010, 2037-2053.
[2] Bollobas, B.: Modern Graph Theory, Graduate Texts in Mathematics. Springer, 1082 New York (1998)
[3] Brylawski, T., Oxley, J.: The Tutte polynomial and its applications. In: White, 1097 N. (ed) Matroid
Applications, Encyclopedia of Mathematics and its Applica- 1098 tions. Cambridge University Press,
Cambridge (1992)
[4] J. Ellis-Monaghan, C. Merino, Graph polynomials and their applications I: the Tutte polynomial, invited
chapter for Structural Analysis of Complex Networks, Matthias Dehmer, ed., Birkhauser, 2010.
[5] J. Ellis-Monaghan, C. Merino, Graph polynomials and their applications II: interrelations and interpretations,
invited chapter for Structural Analysis of Complex Networks, Matthias Dehmer, ed., Birkhauser,
2010.
[6] J. Ellis-Monaghan, I. Moffatt, Graphs on Surfaces: Twisted Duality, Polynomials, and Knots. Invited
monograph for the Springer Briefs Series, submitted
[7] J. Ellis-Monaghan, I. Moffatt, A Penrose polynomial for embedded graphs, in press, European Journal
of Combinatorics.
[8] J. Ellis-Monaghan, G. Pangborn, Using DNA self-assembly design strategies to motivate graph theory
concepts, Math. Model. Nat. Phenom., 6, no. 6 (2011) 96-107.
[9] Sokal, A. D.: Chromatic polynomials, Potts models and all that. Physica A, 1319 279, 324332 (2000)
[10] Sokal, A. D. A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts
models. Markov Process Relat. Fields, 7, 21-38 (2001).
[11] Welsh, D. J. A., Merino, C.: The Potts model and the Tutte polynomial. J. 1304 Math. Phys., 41,
11271152 (2000)
[12] Welsh, D. J. A.: Complexity: Knots, Colorings and Counting, Cambridge University Press, Cambridge
(1993).
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