458 research outputs found
Counting matroids in minor-closed classes
A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an -minor is
asymptotically small in case is one of the sparse paving matroids
, , , , or , thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without -minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure
On minor-closed classes of matroids with exponential growth rate
Let \cM be a minor-closed class of matroids that does not contain
arbitrarily long lines. The growth rate function, h:\bN\rightarrow \bN of
\cM is given by h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$). The
Growth Rate Theorem shows that there is an integer such that either:
, or , or there is a
prime-power such that ; this
separates classes into those of linear density, quadratic density, and base-
exponential density. For classes of base- exponential density that contain
no -point line, we prove that for all
sufficiently large . We also prove that, for classes of base- exponential
density that contain no -point line, there exists k\in\bN such
that for all
sufficiently large
Excluded minors for the class of split matroids
The class of split matroids arises by putting conditions on the system of
split hyperplanes of the matroid base polytope. It can alternatively be defined
in terms of structural properties of the matroid. We use this structural
description to give an excluded minor characterisation of the class
Feynman graph polynomials
The integrand of any multi-loop integral is characterised after Feynman
parametrisation by two polynomials. In this review we summarise the properties
of these polynomials. Topics covered in this article include among others:
Spanning trees and spanning forests, the all-minors matrix-tree theorem,
recursion relations due to contraction and deletion of edges, Dodgson's
identity and matroids.Comment: 35 pages, references adde
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