27,297 research outputs found
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
Notes on higher-dimensional partitions
We show the existence of a series of transforms that capture several
structures that underlie higher-dimensional partitions. These transforms lead
to a sequence of triangles whose entries are given combinatorial
interpretations as the number of particular types of skew Ferrers diagrams. The
end result of our analysis is the existence of a triangle, that we denote by F,
which implies that the data needed to compute the number of partitions of a
given positive integer is reduced by a factor of half. The number of spanning
rooted forests appears intriguingly in a family of entries in the triangle F.
Using modifications of an algorithm due to Bratley-McKay, we are able to
directly enumerate entries in some of the triangles. As a result, we have been
able to compute numbers of partitions of positive integers <= 25 in any
dimension.Comment: 36 pages; Mathematica file attached; See
http://www.physics.iitm.ac.in/~suresh/partitions.html to generate numbers of
partition
Spanning forests and the vector bundle Laplacian
The classical matrix-tree theorem relates the determinant of the
combinatorial Laplacian on a graph to the number of spanning trees. We
generalize this result to Laplacians on one- and two-dimensional vector
bundles, giving a combinatorial interpretation of their determinants in terms
of so-called cycle rooted spanning forests (CRSFs). We construct natural
measures on CRSFs for which the edges form a determinantal process. This theory
gives a natural generalization of the spanning tree process adapted to graphs
embedded on surfaces. We give a number of other applications, for example, we
compute the probability that a loop-erased random walk on a planar graph
between two vertices on the outer boundary passes left of two given faces. This
probability cannot be computed using the standard Laplacian alone.Comment: Published in at http://dx.doi.org/10.1214/10-AOP596 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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