393 research outputs found
Distributive Lattices, Polyhedra, and Generalized Flow
A D-polyhedron is a polyhedron such that if are in then so are
their componentwise max and min. In other words, the point set of a
D-polyhedron forms a distributive lattice with the dominance order. We provide
a full characterization of the bounding hyperplanes of D-polyhedra.
Aside from being a nice combination of geometric and order theoretic
concepts, D-polyhedra are a unifying generalization of several distributive
lattices which arise from graphs. In fact every D-polyhedron corresponds to a
directed graph with arc-parameters, such that every point in the polyhedron
corresponds to a vertex potential on the graph. Alternatively, an edge-based
description of the point set can be given. The objects in this model are dual
to generalized flows, i.e., dual to flows with gains and losses.
These models can be specialized to yield some cases of distributive lattices
that have been studied previously. Particular specializations are: lattices of
flows of planar digraphs (Khuller, Naor and Klein), of -orientations of
planar graphs (Felsner), of c-orientations (Propp) and of -bonds of
digraphs (Felsner and Knauer). As an additional application we exhibit a
distributive lattice structure on generalized flow of breakeven planar
digraphs.Comment: 17 pages, 3 figure
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
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
Cuts and flows of cell complexes
We study the vector spaces and integer lattices of cuts and flows associated
with an arbitrary finite CW complex, and their relationships to group
invariants including the critical group of a complex. Our results extend to
higher dimension the theory of cuts and flows in graphs, most notably the work
of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut
and flow spaces, interpret their coefficients topologically, and give
sufficient conditions for them to be integral bases of the cut and flow
lattices. Second, we determine the precise relationships between the
discriminant groups of the cut and flow lattices and the higher critical and
cocritical groups with error terms corresponding to torsion (co)homology. As an
application, we generalize a result of Kotani and Sunada to give bounds for the
complexity, girth, and connectivity of a complex in terms of Hermite's
constant.Comment: 30 pages. Final version, to appear in Journal of Algebraic
Combinatoric
Positively oriented matroids are realizable
We prove da Silva's 1987 conjecture that any positively oriented matroid is a
positroid; that is, it can be realized by a set of vectors in a real vector
space. It follows from this result and a result of the third author that the
positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a
closed ball.Comment: 20 pages, 3 figures, references adde
Flows on Simplicial Complexes
Given a graph , the number of nowhere-zero \ZZ_q-flows is
known to be a polynomial in . We extend the definition of nowhere-zero
\ZZ_q-flows to simplicial complexes of dimension greater than one,
and prove the polynomiality of the corresponding function
for certain and certain subclasses of simplicial complexes.Comment: 10 pages, to appear in Discrete Mathematics and Theoretical Computer
Science (proceedings of FPSAC'12
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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