Tutte polynomial Research Papers - Academia.edu (original) (raw)

The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded... more

The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by B. Bollobás and O. Riordan.

The Tutte polynomial and the Aharonov-Arab-Ebal-Landau algorithm are applied to Social Network Analysis (SNA) for Epidemiology, Biosurveillance and Biosecurity. We use the methods of Algebraic Computational SNA and of Topological Quantum... more

The Tutte polynomial and the Aharonov-Arab-Ebal-Landau algorithm are applied to Social Network Analysis (SNA) for Epidemiology, Biosurveillance and Biosecurity. We use the methods of Algebraic Computational SNA and of Topological Quantum Computation. The Tutte polynomial is used to describe both the evolution of a social network as the reduced network when some nodes are deleted in an original network and the basic reproductive number for a spatial model with bi-networks, borders and memories. We obtain explicit equations that relate evaluations of the Tutte polynomial with epidemiological parameters such as infectiousness, diffusivity and percolation. We claim, finally, that future topological quantum computers will be very important tools in Epidemiology and that the representation of social networks as ribbon graphs will permit the full application of the Bollobás-Riordan-Tutte polynomial with all its combinatorial universality to be epidemiologically relevant.

The following expository article is intended to describe a correspon- dence between matroids and codes. The key results are that the weight enumerator of a code is a specialisation of the Tutte polynomial of the cor- responding matroid,... more

The following expository article is intended to describe a correspon- dence between matroids and codes. The key results are that the weight enumerator of a code is a specialisation of the Tutte polynomial of the cor- responding matroid, and that the MacWilliams relation between weight enu- merators of a code and its dual can be obtained from matroid duality. It also provides a general introduction to matroids, an introduction to trellis decoding, and an algebraic construction of the minimal trellis of a code. Some of this material was presented in the QMW study group, although this version is my own re-working of it. I am grateful to Carrie Ruther- ford and Costas Papadopoulos for their contributions. Some of Carrie's are acknowledged in the text, while Costas taught me about trellis decoding.

The Jones polynomial of an alternating link is a certain specializa- tion of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollob as{Riordan{Tutte polynomial gener- alizes... more

The Jones polynomial of an alternating link is a certain specializa- tion of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollob as{Riordan{Tutte polynomial gener- alizes the Tutte plolynomial of planar graphs to graphs that are embedded in closed surfaces of higher genus (i.e. dessins d'enfant). In this paper we show

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a... more

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L t and for arbitrarily great... more

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L t and for arbitrarily great length L ℓ, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v)= sumj=1NZ,G,lambdacZ,G,j(lambdaZ,G,j)Lell\sum {_{j = 1}^{N_{Z,G,\lambda } } c_{Z,G,j} (\lambda _{Z,G,j} )} ^{L_\ell}sumj=1NZ,G,lambdacZ,G,j(lambdaZ,G,j)Lell . We give general formulas for N Z, G, j and its specialization to v=−1 for arbitrary L t for both types of boundary conditions, as well as other general structural results on Z. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus BBB , arising as the accumulation set of partition function zeros as L ℓ→∞, in the q plane for fixed v and in the v plane for fixed q.

This paper deals with the location of the complex zeros of the Tutte polynomial for a class of self-dual graphs. For this class of graphs, as the form of the eigenvalues is known, the regions of the complex plane can be focused on the... more

This paper deals with the location of the complex zeros of the Tutte polynomial for a class of self-dual graphs. For this class of graphs, as the form of the eigenvalues is known, the regions of the complex plane can be focused on the sets where there is only one dominant eigenvalue in particular containing the positive half plane. Thus, in these regions, the analyticity of the pressure can be derived easily. Next, some examples of graphs with their Tutte polynomial having a few number of eigenvalues are given. The cases of the strip of triangles with a double edge, the wheel and the cycle with an edge having a high order of multiplicity are presented. In particular, for this last example, we remark that the well known conjecture of Chen et al. is false in the finite case.

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x + y = p lying in the positive quadrant. Every coloopless paving matroid is in the class of matroids which contain two... more

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x + y = p lying in the positive quadrant. Every coloopless paving matroid is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM (a, a) ≤ max{TM (2a, 0), TM (0, 2a)} for a ≥ 2. We conjecture that TM (1, 1) ≤ max{TM (2, 0), TM (0, 2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.

We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this... more

We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

The Tutte polynomial is a powerfull analytic tool to study the structure of planar graphs. In this paper, we establish some relations between the number of clusters per bond for planar graph and its dual : these relations bring into play... more

The Tutte polynomial is a powerfull analytic tool to study the structure of planar graphs. In this paper, we establish some relations between the number of clusters per bond for planar graph and its dual : these relations bring into play the coordination number of the graphs. The factorial moment measure of the number of clusters per bond are given using the derivative of the Tutte polynomial. Examples are presented for simple planar graph. The cases of square, triangular, honeycomb, Archimedean and Laves lattices are discussed.

We present generalisations of several MacWilliams type identities, including those by Klve and Shiromoto, and of the theorems of Greene and Barg that describe support weight enumerators of the code. One of our main tools is a... more

We present generalisations of several MacWilliams type identities, including those by Klve and Shiromoto, and of the theorems of Greene and Barg that describe support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski. 1

We want to explain some formulas appearing in connection with root systems and their zonotopes which are relevant for the theory of the Kostant partition function. In particular, we compute explicitly the Tutte polynomial for all... more

We want to explain some formulas appearing in connection with root systems and their zonotopes which are relevant for the theory of the Kostant partition function. In particular, we compute explicitly the Tutte polynomial for all exceptional root systems. A more systematic treatment of these topics will appear in a forthcoming book Topics in Hyperplane Arrangements, Polytopes and Box-Splines.

Let GGG be a connected graph with vertex set 0,1,2,...,n\{0,1,2,...,n\}0,1,2,...,n. We allow GGG to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of GGG-parking functions. In particular, we... more

Let GGG be a connected graph with vertex set 0,1,2,...,n\{0,1,2,...,n\}0,1,2,...,n. We allow GGG to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of GGG-parking functions. In particular, we give the definition of the bridge vertex of a GGG-parking function and obtain an expression of the Tutte polynomial TG(x,y)T_G(x,y)TG(x,y) of GGG in terms of GGG-parking functions. We find the Tutte polynomial enumerates the GGG-parking function by the number of the bridge vertices.

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and... more

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)= sumj=1NZ,G,lambda\sum _{j = 1}^{N_{Z,G,\lambda } }sumj=1NZ,G,lambda c z,G,j (λ z,G,j )m-1. We give general formulas for N Z,G,j and its specialization to v=−1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus mathcalB{\mathcal{B}}mathcalB , arising as the accumulation set of partition function zeros as m→∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L=3 (free) and L=3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q=3 Potts antiferromagnet on the 2D triangular lattice is given.

A graph polynomial p(G,barX)p(G, \bar{X})p(G,barX) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points barX=barx_0\bar{X}= \bar{x}_0barX=barx_0 . In this paper we... more

A graph polynomial p(G,barX)p(G, \bar{X})p(G,barX) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points barX=barx_0\bar{X}= \bar{x}_0barX=barx_0 . In this paper we study the question how to prove that a given graph parameter, say ω(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials. Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f(G) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL, the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.

This is a continuation of our paper \A Theory of Pfaan Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of T -joins and k-cuts of graphs. As a consequence, we... more

This is a continuation of our paper \A Theory of Pfaan Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of T -joins and k-cuts of graphs. As a consequence, we show that the computational problem to nd the maximum weight of an edge-cut is polynomially solvable for the instances (G;w) where

In a publication by Noy and Ribó, it was shown that recursively constructible families of graphs are recursive. The authors also conjecture that the converse holds; that is, recursive families are also recursively constructible. In this... more

In a publication by Noy and Ribó, it was shown that recursively constructible families of graphs are recursive. The authors also conjecture that the converse holds; that is, recursive families are also recursively constructible. In this paper, we provide two specific counterexamples to this conjecture, which we then extend to an infinite family of counterexamples.

Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning... more

Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of ...