Results 1  10
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12
Specializations of Ferrers ideals
 J. Algebraic Combin
"... Abstract. We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squar ..."
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Abstract. We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squarefree strongly stable ideal whose minimal generators have degree two. In particular, this shows that threshold graphs can be obtained as specializations of Ferrers graphs, which explains their similar properties. 1.
Reconstruction of complete interval tournaments. II.
, 2010
"... Let a, b (b ≥ a) and n (n ≥ 2) be nonnegative integers and let T (a, b, n) be the set of such generalised tournaments, in which every pair of distinct players is connected at most with b, and at least with a arcs. In [40] we gave a necessary and sufficient condition to decide whether a given sequen ..."
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Cited by 7 (3 self)
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Let a, b (b ≥ a) and n (n ≥ 2) be nonnegative integers and let T (a, b, n) be the set of such generalised tournaments, in which every pair of distinct players is connected at most with b, and at least with a arcs. In [40] we gave a necessary and sufficient condition to decide whether a given sequence of nonnegative integers D = (d1, d2,..., dn) can be realized as the outdegree sequence of a T ∈ T (a, b, n). Extending the results of [40] we show that for any sequence of nonnegative integers D there exist f and g such that some element T ∈ T (g, f, n) has D as its outdegree sequence, and for any (a, b, n)tournament T ′ with the same outdegree sequence D hold a ≤ g and b ≥ f. We propose a Θ(n) algorithm to determine f and g and an O(dnn 2) algorithm to construct a corresponding tournament T.
Threshold graphs, shifted complexes, and graphical complexes
 Discrete Math
"... Abstract. We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations am ..."
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Abstract. We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations among them including new characterizations of the class of threshold graphs. 1.
CELLULAR SPANNING TREES AND LAPLACIANS OF CUBICAL COMPLEXES
, 2009
"... We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigen ..."
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We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin’s enumeration of spanning trees of a complete colorful simplicial complex from the cellular MatrixTree Theorem together with a result of Kook, Reiner and Stanton.
RELATIONS ON GENERALIZED DEGREE SEQUENCES
, 2009
"... We study degree sequences for simplicial posets and polyhedral complexes, generalizing the wellstudied graphical degree sequences. Here we extend the more common generalization of vertextofacet degree sequences by considering arbitrary facetoflag degree sequences. In particular, these may be v ..."
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We study degree sequences for simplicial posets and polyhedral complexes, generalizing the wellstudied graphical degree sequences. Here we extend the more common generalization of vertextofacet degree sequences by considering arbitrary facetoflag degree sequences. In particular, these may be viewed as natural refinements of the flag fvector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an fvector inequality on simplicial posets first shown by Stanley.
New Results on Degree Sequences of Uniform Hypergraphs
, 2009
"... Abstract A sequence of nonnegative integers is kgraphic if it is the degree sequence of a kuniform hypergraph. The only known characterization of kgraphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to dete ..."
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Abstract A sequence of nonnegative integers is kgraphic if it is the degree sequence of a kuniform hypergraph. The only known characterization of kgraphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is kgraphic. In light of this, we present sharp sufficient conditions for kgraphicality based on a sequence's length and degree sum. Kocay and Li gave a family of edge exchanges (an extension of 2switches) that could be used to transform one realization of a 3graphic sequence into any other realization. We extend their result to kgraphic sequences for all k 3. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.
REU Report
, 2007
"... We look at a generalized version of degree sequence of hypergraphs, where we consider linear combinations of (hyper)edges with rational, integer or positive integer coefficients, and try to describe the algebraic objects thus defined (QE(G), ZE(G), Z+E(G)) in terms of properties of the hypergraph. ..."
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We look at a generalized version of degree sequence of hypergraphs, where we consider linear combinations of (hyper)edges with rational, integer or positive integer coefficients, and try to describe the algebraic objects thus defined (QE(G), ZE(G), Z+E(G)) in terms of properties of the hypergraph. We find a restrictive definition of connectivity for hypergraphs which ensures a description of ZE(G) in terms of partitions of k into positive integers. We also look at the integral closure of the Z+E(G), which has been fully described for graphs, and give varied examples of why generalizations to higher dimensions encounter problems.