99 research outputs found
On Symmetry of Independence Polynomials
An independent set in a graph is a set of pairwise non-adjacent vertices, and
alpha(G) is the size of a maximum independent set in the graph G. A matching is
a set of non-incident edges, while mu(G) is the cardinality of a maximum
matching.
If s_{k} is the number of independent sets of cardinality k in G, then
I(G;x)=s_{0}+s_{1}x+s_{2}x^{2}+...+s_{\alpha(G)}x^{\alpha(G)} is called the
independence polynomial of G (Gutman and Harary, 1983). If
, 0=< j =< alpha(G), then I(G;x) is called symmetric (or
palindromic). It is known that the graph G*2K_{1} obtained by joining each
vertex of G to two new vertices, has a symmetric independence polynomial
(Stevanovic, 1998). In this paper we show that for every graph G and for each
non-negative integer k =< mu(G), one can build a graph H, such that: G is a
subgraph of H, I(H;x) is symmetric, and I(G*2K_{1};x)=(1+x)^{k}*I(H;x).Comment: 16 pages, 13 figure
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
On the shape of a pure O-sequence
An order ideal is a finite poset X of (monic) monomials such that, whenever M
is in X and N divides M, then N is in X. If all, say t, maximal monomials of X
have the same degree, then X is pure (of type t). A pure O-sequence is the
vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree.
Equivalently, in the language of commutative algebra, pure O-sequences are the
h-vectors of monomial Artinian level algebras. Pure O-sequences had their
origin in one of Richard Stanley's early works in this area, and have since
played a significant role in at least three disciplines: the study of
simplicial complexes and their f-vectors, level algebras, and matroids. This
monograph is intended to be the first systematic study of the theory of pure
O-sequences. Our work, making an extensive use of algebraic and combinatorial
techniques, includes: (i) A characterization of the first half of a pure
O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel;
(ii) A study of (the failing of) the unimodality property; (iii) The problem of
enumerating pure O-sequences, including a proof that almost all O-sequences are
pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type
t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents
perhaps the strongest possible structural result short of an (impossible?)
characterization; (v) A pithy connection of the ICP with Stanley's matroid
h-vector conjecture; (vi) A specific study of pure O-sequences of type 2,
including a proof of the Weak Lefschetz Property in codimension 3 in
characteristic zero. As a corollary, pure O-sequences of codimension 3 and type
2 are unimodal (over any field); (vii) An analysis of the extent to which the
Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii)
Some observations about pure f-vectors, an important special case of pure
O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly
minor revisions with respect to last year's versio
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