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Tutte Polynomial

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The Tutte polynomial of an undirected graph G, also known as the dichromate or Tutte-Whitney polynomial, can be defined from the activities of its spanning trees. Let i denote the cardinal number of the set of internally active edges of a spanning tree T of G, j the cardinal number of the set of externally active edges of T, and t_(ij) the number of spanning trees of G whose internal activity is i and external activity is j. Then

T(x,y)=sumt_(ij)x^iy^j
(1)

(Biggs 1993, p. 100).

An equivalent definition is given by

T(x,y)=sum_(A subset= E)(x-1)^(k_A-k)(y-1)^(|A|-n+k_A),
(2)

where G=(V,E) has n vertices and k connected components, and k_A is the number of connected components of the spanning subgraph (V,A).

Several analogs of the Tutte polynomial have been considered for directed graphs, including the cover polynomial (Chung and Graham 1995), Gordon-Traldi polynomials (Gordon and Traldi 1993), and three-variable B-polynomial (Awan and Bernardi 2016; Chow 2016). However, with the exceptions of the Gordon-Traldi polynomial f_8 and B-polynomial, these are not proper generalizations of the Tutte polynomial since they are not equivalent to the Tutte polynomial for the special case of undirected graphs (Awan and Bernardi 2016).

The Tutte polynomial can be computed in the Wolfram Language using TuttePolynomial[g, {x, y}].

The Tutte polynomial is multiplicative over disjoint unions.

For an undirected graph on n vertices with c connected components, the Tutte polynomial is given by

T(x+1,y+1)=x^(n-c)R(x^(-1),y)
(3)

where R(x,y) is the rank polynomial (generalizing Biggs 1993, p. 101). The Tutte polynomial is therefore a rather general two-variable graph polynomial from which a number of other important one- and two-variable polynomials can be computed.

For not-necessarily connected graphs, the Tutte polynomial T(x,y) is related the chromatic polynomial pi(x), flow polynomial C^*(u), rank polynomial R(x,y), and reliability polynomial C(p) by

pi(x)=(-1)^(n-c)x^cT(1-x,0)
(4)
C^*(u)=(-1)^(m-n+c)T(0,1-u)
(5)
R(x,y)=x^(n-c)T(x^(-1)+1,y+1)
(6)
C(p)=(1-p)^(n-c)p^(m-n+c)T(1,p^(-1)),
(7)

where n is the number of vertices in the graph, m is the number of edges, and c is the number of connected components.

The Tutte polynomial of the dual graph G^* of a graph G is given by

T_(G^*)(x,y)=T_G(y,x),
(8)

i.e., by swapping the variables of the Tutte polynomial of the original graph. A special case of this identity relates the flow polynomial C_G^*(u) of a planar graph G to the chromatic polynomial of its dual graph G^* by

C_G^*(u)=u^(-1)pi_(G^*)(u).
(9)

The Tutte polynomial of a connected graph G is also completely defined by the following two properties (Biggs 1993, p. 103):

1. If e is an edge of G which is neither a loop nor an isthmus, then T_G(x,y)=T(G^((e));x,y)+T(G_((e));x,y).

2. If Lambda_(ij) is formed from a tree with i edges by adding j loops, then T(Lambda_(ij);x,y)=x^iy^j

Closed forms for some special classes of graphs are summarized in the following table, where s=sqrt((1+x+x^2+y)^2-4x^2y) and t=sqrt((x+y+1)^2-4xy). The Tutte polynomial of the web graph was considered by Biggs et al. (1972) and Brennan et al. (2013).

graphT(x,y)
book graph S_(n+1) square P_2((1+x+x^2)^n[x(y-1)-y]+y(x+x^2+y)^n)/(y-1)
centipede graphx^(2n-1)
cycle graph C_n(x^n-x)/(x-1)+y
empty graph K^__n1
forestx^(|E|)
gear graph2^(-n)(-2^n-2^nx-2^ny+2^nxy+(1+x+x^2+y-s)^n+(1+x+x^2+y+s)^n)
helm graphx^n{-1-x-y+xy+2^(-n)[(1-t+x+y)^n+(1+t+x+y)^n]}
ladder graph((s(x-1)+x^3-xy+y+1)(s+x^2+x+y+1)^n-(-sx+s+x^3-xy+y+1)(-s+x^2+x+y+1)^n)/(2^(n+1)sx^2)
ladder rung graphx^n
pan graph(x[x^n+x(y-1)-y])/(x-1)
path graph P_nx^(n-1)
star graph S_nx^(n-1)
sunlet graph C_n circledot K_1(x^(2n)+x^n(x(-1+y)-y))/(x-1)
wheel graph W_nxy-x-y-1+2^(1-n)[(x+y+1+t)^(n-1)+(x+y+1-t)^(n-1)]

The random-cluster model expansion relates the q-state Potts model partition function directly to the Tutte polynomial by

Z_G(q,v)=sum_(A subset= E(G))q^(k(A))v^(|A|)=(x-1)^(k(G))(y-1)^(|V(G)|)T_G(x,y),
(10)

where k(A) is the number of connected components of the spanning subgraph (V(G),A), k(G) is the number of connected components of G, q=(x-1)(y-1), and v=y-1. This fixed-graph identity is distinct from the transfer-matrix expansion for a recursively defined graph family,

T(G_n;x,y)=sum_(j=1)^rc_j(x,y)lambda_j(x,y)^n.
(11)

Here, the dependence on the family parameter n is carried by the eigenvalues lambda_j.

For n>=3, the n-antiprism graph is the cyclic width-2 strip of the triangular lattice considered by Chang and Shrock (2000). Using both deletion-contraction and a transfer matrix, they obtained an exact expansion with six eigenvalues. One is 1, three are the roots of P_3(lambda), and two are the roots of P_2(lambda), where

P_3(lambda)=lambda^3-(3+2x+2y+y^2)lambda^2+[(x+1)^2+2xy(y+1)]lambda-x^2y^2
(12)
P_2(lambda)=lambda^2-[(x+1)^2+y(y+2)]lambda+x^2y^2.
(13)

Consequently, (lambda-1)P_3(lambda)P_2(lambda) is a degree-6 characteristic polynomial, and the antiprism Tutte polynomials satisfy the order-6 recurrence relation recorded below. This exact spectral calculation proves the recurrence order rather than merely fitting it from initial polynomials.

The following table summarizes the recurrence relations for Tutte polynomials for some simple classes of graphs.

graphorderrecurrence
antiprism graph6
book graph S_(n+1) square P_22p_n=(2x^2+2x+y+1)p_(n-1)-(x^2+x+1)(x^2+x+y)p_(n-2)
centipede graph1p_n=x^2p_(n-1)
cycle graph C_n2p_n=(x+1)p_(n-1)-xp_(n-2)
gear graph3p_n=(x^2+x+y+2)p_(n-1)+(-x^2y-x^2-x-y-1)p_(n-2)+x^2yp_(n-3)
helm graph3p_n=x(x+y-2)p_(n-1)-x^2(x+1)(y+1)p_(n-2)+x^4p_(n-3)
ladder graph2p_n=(x^2+x+y+1)p_(n-1)-x^2yp_(n-2)
ladder rung graph1p_n=xp_(n-1)
Möbius ladder M_n6
pan graph2p_n=(x+1)p_(n-1)-xp_(n-2)
path graph1p_n=xp_(n-1)
prism graph Y_n6
star graph S_n1p_n=xp_(n-1)
sunlet graph C_n circledot K_12p_n=x(x+1)p_(n-1)-x^3p_(n-2)
web graph6
wheel graph W_n3p_n=(x+y+2)p_(n-1)-(x+1)(y+1)p_(n-2)+xyp_(n-3)

An equation for the Tutte polynomial T_(K_n)(x,y) of the complete graph K_n was found by Tutte (1954, 1967). In particular, T_(K_n)(x,y) has exponential generating function

sum_(n=1)^inftyT_(K_n)(x,y)(u^n)/(n!)=1/(x-1){[sum_(n=0)^inftyy^((n; 2))(y-1)^(-n)(u^n)/(n!)]^((x-1)(y-1))-1},
(14)

(Gessel 1995, Gessel and Sagan 1996). This can be written more simply in terms of the coboundary polynomial

chi^__G(q,t)=(t-1)^(n_G-c_G)T_G((q+t-1)/(t-1),t),
(15)

where c_G is the connected component count and n_G is the vertex count of a graph G (Martin and Reiner 2005). In this form, the exponential generating function of K_n is given by

1+qsum_(n=1)^inftychi^__(K_n)(q,t)(x^n)/(n!)=(sum_(n=0)^inftyt^((n; 2))(x^n)/(n!))^q,
(16)

which can be converted to the corresponding Tutte polynomial using the above relationship and the substitution q->(x-1)(y-1) and t->y. The formula was rediscovered by Pak in the form of the following recurrence

F_n(x,y)=sum_(k=1)^n(n-1; k-1)(x+y+y^2+...+y^(k-1))F_(k-1)(1,y)F_(n-k)(x,y),
(17)

where F_n(x,y)=T_(K_(n+1))(x,y).

A formula for the Tutte polynomial of a complete bipartite graph K_(m,n) is given in terms of an bivariate exponential generating function for the coboundary polynomial as

1+qsum_(m=0)^inftysum_(n=0)^inftychi^__(K_(m,n))(q,t)(x^my^m)/(m!n!)=(sum_(k=0)^inftysum_(l=0)^inftyt^(kl)(x^ky^l)/(k!l!))^q
(18)

by Martin and Reiner (2005).

Nonisomorphic graphs do not necessarily have distinct Tutte polynomials. de Mier and Noy (2004) call a graph that is determined by its Tutte polynomial a T-unique graph and showed that wheel graphs, ladder graphs, Möbius ladders, complete multipartite graphs (with the exception of T_(1,p)), and hypercube graphs are T-unique graphs. Kuhl (2008) showed that the generalized Petersen graphs GP(m,2) and their line graphs L(GP(m,2)) are T-unique.

The numbers of simple graphs on n=1, 2, ... nodes that are not Tutte-unique for a given value of n are 0, 0, 0, 4, 15, 84, 548, 5629, ... (OEIS A243048), while the corresponding numbers of Tutte-unique graphs are 1, 2, 4, 7, 19, 72, 496, 6717, ... (OEIS A243049). The following table summarizes some small co-Tutte graphs.

nTutte polynomialgraphs
4x^2P_3 union K_1, ladder rung graph 2P_2
4x^3claw graph K_(1,3), path graph P_4
5x^2P_3 union 2K_1, 2P_2 union K_1
5x^3K_(1,3) union K_1, P_3 union P_2, P_4 union P_1
5x^4fork graph, path graph P_5, star graph S_5
5x(x+x^2+y)paw graph union K_1, C_3 union P_2
5x^2(x+x^2+y)bull graph, cricket graph, (3,2)-tadpole graph
5x(x+2x^2+x^3+y+2xy+y^2)dart graph, kite graph

See also

Chromatic Polynomial, Flow Polynomial, Potts Model, Rank Polynomial, Tutte Matrix

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References

Andrzejak, A. "Splitting Formulas for Tutte Polynomials." J. Combin. Th., Ser. B. 70, 346-366, 1997.Andrzejak, A. "An Algorithm for the Tutte Polynomials of Graphs of Bounded Treewidth." Disc. Math. 190, 39-54, 1998.Biggs, N. L. "The Tutte Polynomial." Ch. 13 in Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, pp. 97-105, 1993.Biggs, N. L.; Damerell, R. M.; and Sands, D. A. "Recursive Families of Graphs." J. Combin. Theory Ser. B 12, 123-131, 1972.Björklund, A.; Husfeldt, T.; Kaski, P.; and Koivisto, M. "Computing the Tutte Polynomial in Vertex-Exponential Time." In Proceedings of the IEEE Symposium on the Foundations of Computer Science (FOCS), 677-686, 2008.Brennan, C.; Mansour, T.; and Mphako-Banda, E. "Tutte Polynomials of Wheels Via Generating Functions." Bull. Iranian Math. Soc. 39, 881-891, 2013.Brylawski, T. and Oxley, J. "The Tutte Polynomial and Its Applications." Ch. 6 in Matroid Applications (Ed. N. White). Cambridge, England: Cambridge University Press, pp. 123-225, 1992.Chang, S.-C. and Shrock, R. "Exact Potts Model Partition Function on Strips of the Triangular Lattice." Physica A 286, 189-238, 2000. https://doi.org/10.1016/S0378-4371(00)00225-9.Chow, T Y. "Digraph Analogues of the Tutte Polynomials." Preprint chapter for The CRC Handbook on the Tutte Polynomial and Related Topics (Ed. I. Moffat and J. Ellis-Monaghan). 2016.Chung, F. R. K. and Graham, R. L. "On the Cover Polynomial of a Digraph." J. Combin. Theory, Ser. B 65, 273-290, 1995.de Mier, A. and Noy, M. "On Graphs Determined by Their Tutte Polynomial." Graphs Combin. 20, 105-119, 2004.de Mier, A. and Noy, M. "Tutte Uniqueness of Line Graphs." Disc. Math. 301, 57-65, 2005.Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications I: The Tutte Polynomial." 28 Jun 2008. https://arxiv.org/abs/0803.3079.Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications II: Interrelations and Interpretations." 28 Jun 2008. https://arxiv.org/abs/0806.4699.Gessel, I. M. "Enumerative Applications of a Decomposition for Graphs and Digraphs." Disc. Math. 139, 257-271, 1995.Gessel, I. M. and Sagan, B. E. "The Tutte Polynomial of a Graph, Depth-First Search, and Simplicial Complex Partitions." Elec. J. Combin. 3, No. 2, R9, 1-36, 1996. https://doi.org/10.37236/1267.Gordon, G. and Traldi, L. "Polynomials for Directed Graphs." Congr. Numer. 94, 187-201, 1993.Haggard, G.; Pearce, D. J.; and Royle, G. "Computing Tutte Polynomials" ACM Trans. Math. Software 37, Art. 24, 17 pp., 2010.Jaeger, F. "Tutte Polynomials and Link Polynomials." Proc. Amer. Math. Soc. 103, 647-665, 1988.Jaeger, F.; Vertigan, D.; and Welsh, D. "On the Computational Complexity of the Jones and Tutte Polynomials." Math. Proc. Camb. Phil. Soc. 108, 35-53, 1990.Kuhl, J. S. "The Tutte Polynomial and the Generalized Petersen Graph." Australas. J. Combin. 40, 87-97, 2008.Martin, J. and Reiner, V. "Cyclotomic and Simplicial Matroids." Israel J. Math. 150, 229-240, 2005.Pak, I. "Computation of Tutte Polynomials of Complete Graphs." https://www.math.ucla.edu/~pak/papers/Pak_Computation_Tutte_polynomial_complete_graphs.pdf.Sloane, N. J. A. Sequences A243048 and A243049 in "The On-Line Encyclopedia of Integer Sequences."Tutte, W. T. "A Contribution to the Theory of Chromatic Polynomials." Canad. J. Math. 6, 80-91, 1954.Tutte, W. T. "On Dichromatic Polynomials." J. Combin. Th. 2, 301-320, 1967.

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Tutte Polynomial

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Weisstein, Eric W. "Tutte Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TuttePolynomial.html

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