tutte_polynomial

tutte_polynomial(G)

Returns the Tutte polynomial of G

This function computes the Tutte polynomial via an iterative version of the deletion-contraction algorithm.

The Tutte polynomial T_G(x, y) is a fundamental graph polynomial invariant in two variables. It encodes a wide array of information related to the edge-connectivity of a graph; “Many problems about graphs can be reduced to problems of finding and evaluating the Tutte polynomial at certain values” [1]. In fact, every deletion-contraction-expressible feature of a graph is a specialization of the Tutte polynomial [2] (see Notes for examples).

There are several equivalent definitions; here are three:

Def 1 (rank-nullity expansion): For G an undirected graph, n(G) the number of vertices of G, E the edge set of G, V the vertex set of G, and c(A) the number of connected components of the graph with vertex set V and edge set A [3]:

\[T_G(x, y) = \sum_{A \in E} (x-1)^{c(A) - c(E)} (y-1)^{c(A) + |A| - n(G)}\]

Def 2 (spanning tree expansion): Let G be an undirected graph, T a spanning tree of G, and E the edge set of G. Let E have an arbitrary strict linear order L. Let B_e be the unique minimal nonempty edge cut of \(E \setminus T \cup {e}\). An edge e is internally active with respect to T and L if e is the least edge in B_e according to the linear order L. The internal activity of T (denoted i(T)) is the number of edges in \(E \setminus T\) that are internally active with respect to T and L. Let P_e be the unique path in \(T \cup {e}\) whose source and target vertex are the same. An edge e is externally active with respect to T and L if e is the least edge in P_e according to the linear order L. The external activity of T (denoted e(T)) is the number of edges in \(E \setminus T\) that are externally active with respect to T and L. Then [4] [5]:

\[T_G(x, y) = \sum_{T \text{ a spanning tree of } G} x^{i(T)} y^{e(T)}\]

Def 3 (deletion-contraction recurrence): For G an undirected graph, G-e the graph obtained from G by deleting edge e, G/e the graph obtained from G by contracting edge e, k(G) the number of cut-edges of G, and l(G) the number of self-loops of G:

\[\begin{split}T_G(x, y) = \begin{cases} x^{k(G)} y^{l(G)}, & \text{if all edges are cut-edges or self-loops} \\ T_{G-e}(x, y) + T_{G/e}(x, y), & \text{otherwise, for an arbitrary edge $e$ not a cut-edge or loop} \end{cases}\end{split}\]

Parameters:

GNetworkX graph

Returns:

instance of sympy.core.add.Add

A Sympy expression representing the Tutte polynomial for G.

Notes

Some specializations of the Tutte polynomial:

• T_G(1, 1) counts the number of spanning trees of G

• T_G(1, 2) counts the number of connected spanning subgraphs of G

• T_G(2, 1) counts the number of spanning forests in G

• T_G(0, 2) counts the number of strong orientations of G

• T_G(2, 0) counts the number of acyclic orientations of G

Edge contraction is defined and deletion-contraction is introduced in [6]. Combinatorial meaning of the coefficients is introduced in [7]. Universality, properties, and applications are discussed in [8].

Practically, up-front computation of the Tutte polynomial may be useful when users wish to repeatedly calculate edge-connectivity-related information about one or more graphs.

References

[1]

M. Brandt, “The Tutte Polynomial.” Talking About Combinatorial Objects Seminar, 2015 https://math.berkeley.edu/~brandtm/talks/tutte.pdf

[2]

A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, “Computing the Tutte polynomial in vertex-exponential time” 49th Annual IEEE Symposium on Foundations of Computer Science, 2008 https://ieeexplore.ieee.org/abstract/document/4691000

[3]

Y. Shi, M. Dehmer, X. Li, I. Gutman, “Graph Polynomials,” p. 14

[4]

Y. Shi, M. Dehmer, X. Li, I. Gutman, “Graph Polynomials,” p. 46

[5]

A. Nešetril, J. Goodall, “Graph invariants, homomorphisms, and the Tutte polynomial” https://iuuk.mff.cuni.cz/~andrew/Tutte.pdf

[6]

D. B. West, “Introduction to Graph Theory,” p. 84

[7]

G. Coutinho, “A brief introduction to the Tutte polynomial” Structural Analysis of Complex Networks, 2011 https://homepages.dcc.ufmg.br/~gabriel/seminars/coutinho_tuttepolynomial_seminar.pdf

[8]

J. A. Ellis-Monaghan, C. Merino, “Graph polynomials and their applications I: The Tutte polynomial” Structural Analysis of Complex Networks, 2011 https://arxiv.org/pdf/0803.3079.pdf

Examples

>>> C = nx.cycle_graph(5)
>>> nx.tutte_polynomial(C)
x**4 + x**3 + x**2 + x + y

>>> D = nx.diamond_graph()
>>> nx.tutte_polynomial(D)
x**3 + 2*x**2 + 2*x*y + x + y**2 + y