7 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
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
The MerinoâWelsh conjecture holds for seriesâparallel graphs
The MerinoâWelsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of seriesâparallel graphs