108 research outputs found
A Tutte polynomial inequality for lattice path matroids
Let be a matroid without loops or coloops and let be its Tutte
polynomial. In 1999 Merino and Welsh conjectured that holds for graphic matroids. Ten years later, Conde and
Merino proposed a multiplicative version of the conjecture which implies the
original one. In this paper we prove the multiplicative conjecture for the
family of lattice path matroids (generalizing earlier results on uniform and
Catalan matroids). In order to do this, we introduce and study particular
lattice path matroids, called snakes, used as building bricks to indeed
establish a strengthening of the multiplicative conjecture as well as a
complete characterization of the cases in which equality holds.Comment: 17 pages, 9 figures, improved exposition/minor correction
Some inequalities for the Tutte polynomial
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 matroids is in the class of matroids which contain two
disjoint bases or whose ground set is the union of two bases of M*. For this
latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a
>= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same
class of matroids. We also prove this conjecture for some families of graphs
and matroids.Comment: 17 page
The Merino--Welsh conjecture is false for matroids
The matroidal version of the Merino--Welsh conjecture states that the Tutte
polynomial of any matroid without loops and coloops satisfies
that Equivalently, if the
Merino--Welsh conjecture is true for all matroids without loops and coloops,
then the following inequalities are also satisfied for all matroids without
loops and coloops: and
We show a counter-example for these
inequalities.Comment: 6 page
Order Quasisymmetric Functions Distinguish Rooted Trees
Richard P. Stanley conjectured that finite trees can be distinguished by
their chromatic symmetric functions. In this paper, we prove an analogous
statement for posets: Finite rooted trees can be distinguished by their order
quasisymmetric functions.Comment: 16 pages, 5 figures, referees' suggestions incorporate
On the number of spanning trees in random regular graphs
Let be a fixed integer. We give an asympotic formula for the
expected number of spanning trees in a uniformly random -regular graph with
vertices. (The asymptotics are as , restricted to even if
is odd.) We also obtain the asymptotic distribution of the number of
spanning trees in a uniformly random cubic graph, and conjecture that the
corresponding result holds for arbitrary (fixed) . Numerical evidence is
presented which supports our conjecture.Comment: 26 pages, 1 figure. To appear in the Electronic Journal of
Combinatorics. This version addresses referee's comment
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