1,815 research outputs found
Sparse Hypergraphs and Pebble Game Algorithms
A hypergraph is -sparse if no subset spans
more than hyperedges. We characterize -sparse
hypergraphs in terms of graph theoretic, matroidal and algorithmic properties.
We extend several well-known theorems of Haas, Lov{\'{a}}sz, Nash-Williams,
Tutte, and White and Whiteley, linking arboricity of graphs to certain counts
on the number of edges. We also address the problem of finding
lower-dimensional representations of sparse hypergraphs, and identify a
critical behaviour in terms of the sparsity parameters and . Our
constructions extend the pebble games of Lee and Streinu from graphs to
hypergraphs
Codismantlability and projective dimension of the Stanley-Reisner ring of special hypergraphs
In this paper firstly, we generalize the concept of codismantlable graphs to
hypergraphs and show that some special vertex decomposable hypergraphs are
codismantlable. Then we generalize the concept of bouquet in graphs to
hypergraphs to extend some combinatorial invariants of graphs about
disjointness of a set of bouquets. We use these invariants to characterize the
projective dimension of Stanley-Reisner ring of special hypergraphs in some
sense.Comment: To appear in Proceedings Mathematical Science
The -spectrum of a generalized power hypergraph
The generalized power of a simple graph , denoted by , is
obtained from by blowing up each vertex into an -set and each edge into
a -set, where . When ,
is always odd-bipartite. It is known that is
non-odd-bipartite if and only if is non-bipartite, and
has the same adjacency (respectively, signless Laplacian) spectral radius as
. In this paper, we prove that, regardless of multiplicities, the
-spectrum of \A(G^{k,\frac{k}{2}}) (respectively, \Q(G^{k,\frac{k}{2}}))
consists of all eigenvalues of the adjacency matrices (respectively, the
signless Laplacian matrices) of the connected induced subgraphs (respectively,
modified induced subgraphs) of . As a corollary, has the
same least adjacency (respectively, least signless Laplacian) -eigenvalue as
. We also discuss the limit points of the least adjacency -eigenvalues of
hypergraphs, and construct a sequence of non-odd-bipartite hypergraphs whose
least adjacency -eigenvalues converge to .Comment: arXiv admin note: text overlap with arXiv:1408.330
Lattices and Hypergraphs associated to square-free monomial ideals
Given a square-free monomial ideal in a polynomial ring over a field
, one can associate it with its LCM-lattice and its hypergraph. In
this short note, we establish the connection between the LCM-lattice and the
hypergraph, and in doing so we provide a sufficient condition for removing
higher dimension edges of the hypergraph without impacting the projective
dimension of the square-free monomial ideal. We also offer algorithms to
compute the projective dimension of a class of square-free monomial ideals
built using the new result and previous results of Lin-Mantero.Comment: The paper is split into two via the suggestion of the editor. The
second part will be submitted soo
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
Let be a weakly irreducible nonnegative tensor with spectral
radius . Let (respectively,
) be the set of normalized diagonal matrices arising from
the eigenvectors of corresponding to the eigenvalues with modulus
(respectively, the eigenvalue ). It is
shown that is an abelian group containing
as a subgroup, which acts transitively on the set , where
and is the
stabilizer of . The spectral symmetry of is
characterized by the group , and
is called spectral -symmetric. We obtain the structural information of
by analyzing the property of , especially for
connected hypergraphs we get some results on the edge distribution and
coloring. If moreover is symmetric, we prove that
is spectral -symmetric if and only if it is -colorable. We
characterize the spectral -symmetry of a tensor by using its generalized
traces, and show that for an arbitrarily given integer and each
positive integer with , there always exists an -uniform
hypergraph such that is spectral -symmetric
Quantum walks on regular uniform hypergraphs
Quantum walks on graphs have shown prioritized benefits and applications in
wide areas. In some scenarios, however, it may be more natural and accurate to
mandate high-order relationships for hypergraphs, due to the density of
information stored inherently. Therefore, we can explore the potential of
quantum walks on hypergraphs. In this paper, by presenting the one-to-one
correspondence between regular uniform hypergraphs and bipartite graphs, we
construct a model for quantum walks on bipartite graphs of regular uniform
hypergraphs with Szegedy's quantum walks, which gives rise to a quadratic
speed-up. Furthermore, we deliver spectral properties of the transition matrix,
given that the cardinalities of the two disjoint sets are different in the
bipartite graph. Our model provides the foundation for building quantum
algorithms on the strength of quantum walks, suah as quantum walks search,
quantized Google's PageRank and quantum machine learning, based on hypergraphs.Comment: 16 pages, 1 figure
Hypergraphs and the Regularity of Square-free Monomial Ideals
We define a new combinatorial object, which we call a labeled hypergraph,
uniquely associated to any square-free monomial ideal. We prove several upper
bounds on the regularity of a square-free monomial ideal in terms of simple
combinatorial properties of its labeled hypergraph. We also give specific
formulas for the regularity of square-free monomial ideals with certain labeled
hypergraphs. Furthermore, we prove results in the case of one-dimensional
labeled hypergraphs.Comment: 14 pages (Updated references. Fixed minor typos.
On Computing Maximal Independent Sets of Hypergraphs in Parallel
Whether or not the problem of finding maximal independent sets (MIS) in
hypergraphs is in (R)NC is one of the fundamental problems in the theory of
parallel computing. Unlike the well-understood case of MIS in graphs, for the
hypergraph problem, our knowledge is quite limited despite considerable work.
It is known that the problem is in \emph{RNC} when the edges of the hypergraph
have constant size. For general hypergraphs with vertices and edges,
the fastest previously known algorithm works in time with
processors. In this paper we give an EREW PRAM algorithm
that works in time with processors on general
hypergraphs satisfying , where
and . Our algorithm is
based on a sampling idea that reduces the dimension of the hypergraph and
employs the algorithm for constant dimension hypergraphs as a subroutine
E-cospectral hypergraphs and some hypergraphs determined by their spectra
Two -uniform hypergraphs are said to be cospectral (E-cospectral), if
their adjacency tensors have the same characteristic polynomial
(E-characteristic polynomial). A -uniform hypergraph is said to be
determined by its spectrum, if there is no other non-isomorphic -uniform
hypergraph cospectral with . In this note, we give a method for constructing
E-cospectral hypergraphs, which is similar with Godsil-McKay switching. Some
hypergraphs are shown to be determined by their spectra
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