19 research outputs found
The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities
We show that in an ordered matroid the partial derivative
\partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the
generating function of activities of subsets with corank p and nullity q. More
generally, this property holds for the 3-variable Tutte polynomial of a matroid
perspective.Comment: 28 pages, 3 figures, 5 table
The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations
We show that the 4-variable generating function of certain orientation
related parameters of an ordered oriented matroid is the evaluation at (x + u,
y+v) of its Tutte polynomial. This evaluation contains as special cases the
counting of regions in hyperplane arrangements and of acyclic orientations in
graphs. Several new 2-variable expansions of the Tutte polynomial of an
oriented matroid follow as corollaries.
This result hold more generally for oriented matroid perspectives, with
specific special cases the counting of bounded regions in hyperplane
arrangements or of bipolar acyclic orientations in graphs.
In corollary, we obtain expressions for the partial derivatives of the Tutte
polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table
K-theoretic Tutte polynomials of morphisms of matroids
We generalize the Tutte polynomial of a matroid to a morphism of matroids via
the K-theory of flag varieties. We introduce two different generalizations, and
demonstrate that each has its own merits, where the trade-off is between the
ease of combinatorics and geometry. One generalization recovers the Las Vergnas
Tutte polynomial of a morphism of matroids, which admits a corank-nullity
formula and a deletion-contraction recursion. The other generalization does
not, but better reflects the geometry of flag varieties.Comment: 27 pages; minor revisions. To appear in JCT
Fourientation activities and the Tutte polynomial
International audienceA fourientation of a graph G is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of G which allow for a new 12 variable expansion of the Tutte polynomial TG. Our formula specializes to both an orientation activities expansion of TG due to Las Vergnas (1984) and a generalized activities expansion of TG due to Gordon and Traldi (1990)
The Las Vergnas Polynomial for embedded graphs
The Las Vergnas polynomial is an extension of the Tutte polynomial to
cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as
special case of his Tutte polynomial of a morphism of matroids. While the
general Tutte polynomial of a morphism of matroids has a complete set of
deletion-contraction relations, its specialisation to cellularly embedded
graphs does not. Here we extend the Las Vergnas polynomial to graphs in
pseudo-surfaces. We show that in this setting we can define deletion and
contraction for embedded graphs consistently with the deletion and contraction
of the underlying matroid perspective, thus yielding a version of the Las
Vergnas polynomial with complete recursive definition. This also enables us to
obtain a deeper understanding of the relationships among the Las Vergnas
polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial. We
also take this opportunity to extend some of Las Vergnas' results on Eulerian
circuits from graphs in surfaces of low genus to surfaces of arbitrary genus