36 research outputs found
Shellable graphs and sequentially Cohen-Macaulay bipartite graphs
Associated to a simple undirected graph G is a simplicial complex whose faces
correspond to the independent sets of G. We call a graph G shellable if this
simplicial complex is a shellable simplicial complex in the non-pure sense of
Bjorner-Wachs. We are then interested in determining what families of graphs
have the property that G is shellable. We show that all chordal graphs are
shellable. Furthermore, we classify all the shellable bipartite graphs; they
are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give an
recursive procedure to verify if a bipartite graph is shellable. Because
shellable implies that the associated Stanley-Reisner ring is sequentially
Cohen-Macaulay, our results complement and extend recent work on the problem of
determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We
also give a new proof for a result of Faridi on the sequentially
Cohen-Macaulayness of simplicial forests.Comment: 16 pages; more detail added to some proofs; Corollary 2.10 was been
clarified; the beginning of Section 4 has been rewritten; references updated;
to appear in J. Combin. Theory, Ser.
A new graph invariant arises in toric topology
In this paper, we introduce new combinatorial invariants of any finite simple
graph, which arise in toric topology. We compute the -th (rational) Betti
number and Euler characteristic of the real toric variety associated to a graph
associahedron P_{\B(G)}. They can be calculated by a purely combinatorial
method (in terms of graphs) and are named and , respectively. To
our surprise, for specific families of the graph , our invariants are deeply
related to well-known combinatorial sequences such as the Catalan numbers and
Euler zigzag numbers.Comment: 21 pages, 3 figures, 4 table
Pseudometrics, The Complex of Ultrametrics, and Iterated Cycle Structures
Every set X, finite of cardinality n say, carries a set M(X) of all possible pseudometrics. It is well known that M(X) forms a convex polyhedral cone whose faces correspond to triangle inequalities. Every point in a convex cone can be expressed as a conical sum of its extreme rays, hence the interest around discovering and classifying such rays. We shall give examples of extreme rays for M(X) exhibiting all integral edge lengths up to half the cardinality of X. By intersecting the cone with the unit cube we obtain the convex polytope of bounded-by-one pseudometrics BM(X). Analogous to extreme rays, every point in a convex polytope arises as a convex combination of extreme points. Extreme rays of BM(X) give rise to very special extreme points of ÌBM(X) as we may normalize a nonzero pseudometric to make its largest distance 1. We shall give a simple and complete characterization of extremeness for metrics with only edge lengths equal to 1/2 and 1. Then we shall use this characterization to give a decomposition result for the upper half of BM(X). BM(X) contains the set of bounded-by-1 pseudoultrametrics, U(X). Ultrametrics satisfy a stronger version of the triangle inequality, and have an interesting structure expressed in terms of partition chains. We will describe the topology of U(X) and its subset of scaled ultrametrics, SU(X), up to homotopy equivalence. Every permutation on a set X can be written as a product of disjoint cycles that cover X. In this way, a permutation generalizes a partition. An iterated cycle structure (ICS) will then be the associated generalization of a partition chain. Analogously, we will compute the âEuler-characteristicâ of the set of iterated cycle structures
Topological and Geometric Combinatorics
[no abstract available