145 research outputs found
Bounds on the Coefficients of Tension and Flow Polynomials
The goal of this article is to obtain bounds on the coefficients of modular
and integral flow and tension polynomials of graphs. To this end we make use of
the fact that these polynomials can be realized as Ehrhart polynomials of
inside-out polytopes. Inside-out polytopes come with an associated relative
polytopal complex and, for a wide class of inside-out polytopes, we show that
this complex has a convex ear decomposition. This leads to the desired bounds
on the coefficients of these polynomials.Comment: 16 page
One-Point Suspensions and Wreath Products of Polytopes and Spheres
It is known that the suspension of a simplicial complex can be realized with
only one additional point. Suitable iterations of this construction generate
highly symmetric simplicial complexes with various interesting combinatorial
and topological properties. In particular, infinitely many non-PL spheres as
well as contractible simplicial complexes with a vertex-transitive group of
automorphisms can be obtained in this way.Comment: 17 pages, 8 figure
The geometry and combinatorics of cographic toric face rings
In this paper we define and study a ring associated to a graph that we call
the cographic toric face ring, or simply the cographic ring. The cographic ring
is the toric face ring defined by the following equivalent combinatorial
structures of a graph: the cographic arrangement of hyperplanes, the Voronoi
polytope, and the poset of totally cyclic orientations. We describe the
properties of the cographic ring and, in particular, relate the invariants of
the ring to the invariants of the corresponding graph. Our study of the
cographic ring fits into a body of work on describing rings constructed from
graphs. Among the rings that can be constructed from a graph, cographic rings
are particularly interesting because they appear in the study of compactified
Jacobians of nodal curves.Comment: 27 pages; final version, to appear in Algebra & Number Theor
Virtual polytopes
Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes
represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex
polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Virtual Polytopes
Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications
- …