104 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
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
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
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