1,305 research outputs found
Graph products of spheres, associative graded algebras and Hilbert series
Given a finite, simple, vertex-weighted graph, we construct a graded
associative (non-commutative) algebra, whose generators correspond to vertices
and whose ideal of relations has generators that are graded commutators
corresponding to edges. We show that the Hilbert series of this algebra is the
inverse of the clique polynomial of the graph. Using this result it easy to
recognize if the ideal is inert, from which strong results on the algebra
follow. Non-commutative Grobner bases play an important role in our proof.
There is an interesting application to toric topology. This algebra arises
naturally from a partial product of spheres, which is a special case of a
generalized moment-angle complex. We apply our result to the loop-space
homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more
citations, to appear in Mathematische Zeitschrif
Betti numbers of Stanley--Reisner rings with pure resolutions
Let be simplicial complex and let denote the
Stanley--Reisner ring corresponding to . Suppose that has a
pure free resolution. Then we describe the Betti numbers and the
Hilbert--Samuel multiplicity of in terms of the --vector of
. As an application, we derive a linear equation system and some
inequalities for the components of the --vector of the clique complex of an
arbitrary chordal graph. As an other application, we derive a linear equation
system and some inequalities for the components of the --vector of
Cohen--Macaulay simplicial complexes.Comment: 18 pages, better introduction, ask for feedback before submissio
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
Ehrhart clutters: Regularity and Max-Flow Min-Cut
If C is a clutter with n vertices and q edges whose clutter matrix has column
vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a
Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to
show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then
C is an Ehrhart clutter and in this case we provide sharp bounds on the
Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols
conjecture on packing problems, we conjecture that if C is both ideal and the
clique clutter of a perfect graph, then C has the MFMC property. We prove this
conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel
graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof
of our conjecture when C is a uniform clique clutter of a perfect graph. We
close with a generalization of Ehrhart clutters as it relates to total dual
integrality.Comment: Electronic Journal of Combinatorics, to appea
Cohen-Macaulay binomial edge ideals
We study the depth of classes of binomial edge ideals and classify all closed
graphs whose binomial edge ideal is Cohen--Macaulay.Comment: 9 page
Determinantal Facet Ideals
We consider ideals generated by general sets of -minors of an -matrix of indeterminates. The generators are identified with the facets of
an -dimensional pure simplicial complex. The ideal generated by the
minors corresponding to the facets of such a complex is called a determinantal
facet ideal. Given a pure simplicial complex , we discuss the question
when the generating minors of its determinantal facet ideal form a
Gr\"obner basis and when is a prime ideal
Prime splittings of Determinantal Ideals
We consider determinantal ideals, where the generating minors are encoded in
a hypergraph. We study when the generating minors form a Gr\"obner basis. In
this case, the ideal is radical, and we can describe algebraic and numerical
invariants of these ideals in terms of combinatorial data of their hypergraphs,
such as the clique decomposition. In particular, we can construct a minimal
free resolution as a tensor product of the minimal free resolution of their
cliques. For several classes of hypergraphs we find a combinatorial description
of the minimal primes in terms of a prime splitting. That is, we write the
determinantal ideal as a sum of smaller determinantal ideals such that each
minimal prime is a sum of minimal primes of the summands.Comment: Final version to appear in Communications in Algebr
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