14 research outputs found
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
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric
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
Simplicial matrix-tree theorems
We generalize the definition and enumeration of spanning trees from the
setting of graphs to that of arbitrary-dimensional simplicial complexes
, extending an idea due to G. Kalai. We prove a simplicial version of
the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the
squares of the orders of their top-dimensional integral homology groups, in
terms of the Laplacian matrix of . As in the graphic case, one can
obtain a more finely weighted generating function for simplicial spanning trees
by assigning an indeterminate to each vertex of and replacing the
entries of the Laplacian with Laurent monomials. When is a shifted
complex, we give a combinatorial interpretation of the eigenvalues of its
weighted Laplacian and prove that they determine its set of faces uniquely,
generalizing known results about threshold graphs and unweighted Laplacian
eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math.
So
Simplicial effective resistance and enumeration of spanning trees
A graph can be regarded as an electrical network in which each edge is a
resistor. This point of view relates combinatorial quantities, such as the
number of spanning trees, to electrical ones such as effective resistance. The
second and third authors have extended the combinatorics/electricity analogy to
higher dimension and expressed the simplicial analogue of effective resistance
as a ratio of weighted tree enumerators. In this paper, we first use that ratio
to prove a new enumeration formula for color-shifted complexes, confirming a
conjecture by Aalipour and the first author, and generalizing a result of
Ehrenborg and van Willigenburg on Ferrers graphs. We then use the same
technique to recover an enumeration formula for shifted complexes, first proved
by Klivans and the first and fourth authors. In each case, we add facets one at
a time, and give explicit expressions for simplicial effective resistances of
added facets by constructing high-dimensional analogues of currents and
voltages (respectively homological cycles and cohomological cocycles).Comment: 27 pages, minor revisions from v