共查询到20条相似文献,搜索用时 31 毫秒
1.
Christos A. Athanasiadis 《Arkiv f?r Matematik》2011,49(1):17-29
A simplicial complex Δ is called flag if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold
of dimension d−1, then the graph of Δ (i) is (2d−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the d-dimensional cross-polytope. Second, the h-vector of a flag simplicial homology sphere Δ of dimension d−1 is minimized when Δ is the boundary complex of the d-dimensional cross-polytope. 相似文献
2.
Shmuel Onn 《European Journal of Combinatorics》1997,18(8):921-938
The class ofStrongly Signablepartially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs' recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flagh-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable posets, polyhedral fans, and face lattices of partitionable simplicial complexes are all strongly signable, and it is conjectured that all spherical posets are. It is concluded that the barycentric subdivision of a partitionable complex is again partitionable, and an algorithm for producing a partitioning of the subdivision from a partitioning of the complex is described. An expression for the flagh-polynomial of a simplicial complex in terms of itsh-vector is given, and is used to demonstrate that the flagh-vector is symmetric or non-negative whenever theh-vector is. 相似文献
3.
We prove that the γ-vector of the barycentric subdivision of a simplicial sphere is the f-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex. 相似文献
4.
Neeta Pandey 《Proceedings Mathematical Sciences》1999,109(1):1-10
We define a class of simplicial maps — those which are “expanding directions preserving” — from a barycentric subdivision to the original simplicial complex. These maps naturally induce a self map on the links
of their fixed points. The local index at a fixed point of such a map turns out to be the Lefschetz number of the induced
map on the link of the fixed point in relative homology. We also show that a weakly hyperbolic [4] simplicial map sdnK →K is expanding directions preserving. 相似文献
5.
The face numbers of simplicial complexes without missing faces of dimension larger than i are studied. It is shown that among all such (d−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal f-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal h-vector. It is also verified that the l-skeleton of a flag (d−1)-dimensional 2-CM complex is 2(d−l)-CM, while the l-skeleton of a flag piecewise linear (d−1)-sphere is 2(d−l)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension
and the number of vertices are established. 相似文献
6.
Sarfraz Ahmad 《Czechoslovak Mathematical Journal》2013,63(4):989-994
For a simplicial complex Δ we study the behavior of its f- and h-triangle under the action of barycentric subdivision. In particular we describe the f- and h-triangle of its barycentric subdivision sd(Δ). The same has been done for f- and h-vector of sd(Δ) by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the h-triangle of Δ are nonnegative, then the entries of the h-triangle of sd(Δ) are also nonnegative. We conclude with a few properties of the h-triangle of sd(Δ). 相似文献
7.
Sam Payne 《Discrete and Computational Geometry》2008,40(3):365-376
We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms
of multivariate analogues of the h-polynomials of the subdivision and “local contributions” of the links of its nonunimodular faces. We also compute new examples
of nonunimodal h
*-vectors of reflexive polytopes.
Supported by the Clay Mathematics Institute. 相似文献
8.
Bin Zhu 《Journal of Algebraic Combinatorics》2008,27(1):35-54
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using
d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations
of valued quivers, we define a d-compatibility degree (−∥−) on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case
and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost
positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is
the generalized cluster complex defined by Fomin and Reading.
Supported by the NSF of China (Grants 10471071) and by the Leverhulme Trust through the network ‘Algebras, Representations
and Applications’. 相似文献
9.
A simplicial complex
K\mathsf{K}
is called d
-representable if it is the nerve of a collection of convex sets in ℝ
d
;
K\mathsf{K}
is d
-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d−1 that is contained in a unique maximal face; and
K\mathsf{K}
is d
-Leray if every induced subcomplex of
K\mathsf{K}
has vanishing homology of dimension d and larger.
It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d≥2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results, “d-representable” in the assumption can be replaced by “d-collapsible” or even “d-Leray.” 相似文献
10.
G. Hetyei 《Discrete and Computational Geometry》1995,14(1):305-330
We investigate the properties of the Stanley ring of a cubical complex, a cubical analogue of the Stanley-Reisner ring of
a simplicial complex. We compute its Hilbert series in terms of thef-vector, and prove that by taking the initial ideal of the defining relations, with respect to the reverse lexicographic order,
we obtain the defining relations of the Stanley-Reisner ring of the triangulation via “pulling the vertices” of the cubical
complex. Applying an old idea of Hochster we see that this ring is Cohen-Macaulay when the complex is shellable, and we show
that its defining ideal is generated by quadrics when the complex is also a subcomplex of the boundary complex of a convex
cubical polytope. We present a cubical analogue of balanced Cohen-Macaulay simplicial complexes: the class of edge-orientable
shellable cubical complexes. Using Stanley's results about balanced Cohen-Macaulay simplicial complexes and the degree two
homogeneous generating system of the defining ideal, we obtain an infinite set of examples for a conjecture of Eisenbud, Green,
and Harris. This conjecture says that theh-vector of a polynomial ring inn variables modulo an ideal which has ann-element homogeneous system of parameters of degree two, is thef-vector of a simplicial complex. 相似文献
11.
Carl W. Lee 《Israel Journal of Mathematics》1984,47(4):261-269
Letf(P
s
d
) be the set of allf-vectors of simpliciald-polytopes. ForP a simplicial 2d-polytope let Σ(P) denote the boundary complex ofP. We show that for eachf ∈f(P
s
d
) there is a simpliciald-polytopeP withf(P)=f such that the 11 02 simplicial diameter of Σ(P) is no more thanf
0(P)−d+1 (one greater than the conjectured Hirsch bound) and thatP admits a subdivision into a simpliciald-ball with no new vertices that satisfies the Hirsch property. Further, we demonstrate that the number of bistellar operations
required to obtain Σ(P) from the boundary of ad-simplex is minimum over the class of all simplicial polytopes with the samef-vector. This polytopeP will be the one constructed to prove the sufficiency of McMullen's conditions forf-vectors of simplicial polytopes. 相似文献
12.
Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all
Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short
simplicial h -vector. 相似文献
13.
We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including
Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct
explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another
direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement
of Gal’s conjecture, which asserts that such γ-vectors are nonnegative. 相似文献
14.
Christos A. Athanasiadis 《Annals of Combinatorics》2012,16(3):421-448
Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local h-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the h-vector of a simplicial complex is played by the (short or long) cubical h-vector of a cubical complex, defined by Adin, and the role of the local h-vector of a triangulation of a simplex is played by the (short or long) cubical local h-vector of a cubical subdivision of a cube. The cubical local h-vectors are defined in this paper and are shown to share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed. 相似文献
15.
Gunnar Fløystad 《Journal of Algebraic Combinatorics》2007,25(3):285-307
For a simplicial complex Δ on {1, 2,…, n} we define enriched homology and cohomology modules. They are graded modules over k[x
1,…, x
n
] whose ranks are equal to the dimensions of the reduced homology and cohomology groups.
We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein* complexes Δ, and also orientable homology manifolds in terms of the enriched modules. We introduce the notion of girth for
simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex.
We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get.
They are block designs and include Steiner systems S(c, d, n) and cyclic polytopes of even dimension.
This paper is to a large extent a complete rewriting of a previous preprint, “Hierarchies of simplicial complexes via the
BGG-correspondence”. Also Propositions 1.7 and 3.1 have been generalized to cell complexes in [11]. 相似文献
16.
Jason Bell 《Discrete Mathematics》2007,307(6):668-682
We show that each polynomial a(z)=1+a1z+?+adzd in N[z] having only real zeros is the f-polynomial of a multicomplex. It follows that a(z) is also the h-polynomial of a Cohen-Macaulay ring and is the g-polynomial of a simplicial polytope. We conjecture that a(z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zeros of a(z) belong to the real interval [-1,0). We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form. 相似文献
17.
Given the f-vector f = (f0, f1, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δf with f(Δf) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has
for all i and j, where f(Δ) is the f-vector of Δ and where β
ij
(I
Δ) are graded Betti numbers of the Stanley–Reisner ideal I
Δ of Δ.
The first author is supported by JSPS Research Fellowships for Young Scientists.
Received: 23 January 2006 相似文献
18.
A. A. Gaifullin 《Proceedings of the Steklov Institute of Mathematics》2009,266(1):29-48
We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP2. The automorphism group of X is isomorphic to S
4 × S
3. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP2 admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision $
\bar X
$
\bar X
of the triangulation X such that the classical moment mapping μ: ℂP2 → Δ2 is a simplicial mapping of the triangulation $
\bar X
$
\bar X
onto the barycentric subdivision of the triangle Δ2. We study the relationship of the triangulation X with complex crystallographic groups. 相似文献
19.
Journal of Algebraic Combinatorics - We show that the $$\gamma $$ -vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In... 相似文献
20.
Tiziana Calamoneri Emanuele G. Fusco Richard B. Tan Paola Vocca 《Mathematical Methods of Operations Research》2009,69(2):307-321
An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, . . . , λ} to the nodes of the graph such
that adjacent nodes are assigned integers of at least distance h ≥ 1 apart and all nodes of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λ
h, 1, 1 and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n
3), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give
a linear time approximation algorithm for computing the more general L(h, 1, 1)-labeling for outerplanar graphs that is within additive constants of the optimum values.
This research is partially supported by the European Research Project Algorithmic Principles for Building Efficient Overlay Computers (AEOLUS) and was done during the visit of Richard B. Tan at the Department of Computer Science, University of Rome “Sapienza”, supported
by a visiting fellowship from the University of Rome “Sapienza”. 相似文献