We present a new edge selection heuristic and vertex ordering heuristic that together enable one to compute the Tutte polynomial of much larger sparse graphs than was previously doable. As a specific example, we are able to compute the Tutte polynomial of the truncated icosahedron graph using our Maple implementation in under 15 minutes on a single CPU. This compares with a recent result of Hag...

The multivariate Tutte polynomial ẐM of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable ve to each element e of the ground set E. It encodes the full structure of M . Let v = {ve}e∈E , let K be an arbitrary field, and suppose M is connected. We show that ẐM is irreducible over K(v), and give three self-contained proofs that the Gal...

Fix two lattice paths P and Q from ð0; 0Þ to ðm; rÞ that use East and North steps with P never going above Q: We show that the lattice paths that go from ð0; 0Þ to ðm; rÞ and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroid...

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a code group, the matroid differs only trivially from that which arises from the code. ...

The identity linking the Tutte polynomial with the Potts model on a graph implies the existence of a decomposition resembling that previously obtained for the chromatic polynomial. Specifically, let {Gn} be a family of bracelets in which the base graph has b vertices. Then the Tutte polynomial of Gn can be written as a sum of terms, one for each partition π of a non-negative integer ` ≤ b: (x− ...

The Tutte polynomial TG(X,Y ) of a graph G is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, TG(X,Y ) = TG∗(Y,X) where G ∗ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial com...

Through a series of papers in the 1980’s, Bouchet introduced isotropic systems and the Tutte-Martin polynomial of an isotropic system. Then, Arratia, Bollobás, and Sorkin developed the interlace polynomial of a graph in [ABS00] in response to a DNA sequencing application. The interlace polynomial has generated considerable recent attention, with new results including realizing the original inte...

Las Vergnas’ generalizations of the Tutte polynomial are studied as follows. The theory of Tutte-Grothendieck matroid invariantsfis modified so the Tutte decomposition ,f(M) =f(M\e) +f( M/e) is applied only when e $ P (and e is neither a loop nor an isthmus) where P is a distinguished set of points called ports. The resulting “P-ported” Tutte polynomial tp has variables I, w; q,, qZ, . . . . qm...