1,378 research outputs found
Man and machine thinking about the smooth 4-dimensional Poincar\'e conjecture
While topologists have had possession of possible counterexamples to the
smooth 4-dimensional Poincar\'{e} conjecture (SPC4) for over 30 years, until
recently no invariant has existed which could potentially distinguish these
examples from the standard 4-sphere. Rasmussen's s-invariant, a slice
obstruction within the general framework of Khovanov homology, changes this
state of affairs. We studied a class of knots K for which nonzero s(K) would
yield a counterexample to SPC4. Computations are extremely costly and we had
only completed two tests for those K, with the computations showing that s was
0, when a landmark posting of Akbulut (arXiv:0907.0136) altered the terrain.
His posting, appearing only six days after our initial posting, proved that the
family of ``Cappell--Shaneson'' homotopy spheres that we had geared up to study
were in fact all standard. The method we describe remains viable but will have
to be applied to other examples. Akbulut's work makes SPC4 seem more plausible,
and in another section of this paper we explain that SPC4 is equivalent to an
appropriate generalization of Property R (``in S^3, only an unknot can yield
S^1 x S^2 under surgery''). We hope that this observation, and the rich
relations between Property R and ideas such as taut foliations, contact
geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to
look at SPC4.Comment: 37 pages; changes reflecting that the integer family of
Cappell-Shaneson spheres are now known to be standard (arXiv:0907.0136
Eigenvalue interlacing and weight parameters of graphs
Eigenvalue interlacing is a versatile technique for deriving results in
algebraic combinatorics. In particular, it has been successfully used for
proving a number of results about the relation between the (adjacency matrix or
Laplacian) spectrum of a graph and some of its properties. For instance, some
characterizations of regular partitions, and bounds for some parameters, such
as the independence and chromatic numbers, the diameter, the bandwidth, etc.,
have been obtained. For each parameter of a graph involving the cardinality of
some vertex sets, we can define its corresponding weight parameter by giving
some "weights" (that is, the entries of the positive eigenvector) to the
vertices and replacing cardinalities by square norms. The key point is that
such weights "regularize" the graph, and hence allow us to define a kind of
regular partition, called "pseudo-regular," intended for general graphs. Here
we show how to use interlacing for proving results about some weight parameters
and pseudo-regular partitions of a graph. For instance, generalizing a
well-known result of Lov\'asz, it is shown that the weight Shannon capacity
of a connected graph \G, with vertices and (adjacency matrix)
eigenvalues , satisfies \Theta\le
\Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where
is the (standard) Shannon capacity and \vecnu is the positive
eigenvector normalized to have smallest entry 1. In the special case of regular
graphs, the results obtained have some interesting corollaries, such as an
upper bound for some of the multiplicities of the eigenvalues of a
distance-regular graph. Finally, some results involving the Laplacian spectrum
are derived. spectrum are derived
Generalization of neighborhood complexes
We introduce the notion of r-neighborhood complex for a positive integer r,
which is a natural generalization of Lovasz neighborhood complex. The
topologies of these complexes give some obstructions of the existence of graph
maps. We applied these complexes to prove the nonexistence of graph maps about
Kneser graphs. We prove that the fundamental groups of r-neighborhood complexes
are closely related to the (2r)-fundamental groups defined in the author's
previous paper.Comment: 8 page
Some spectral and quasi-spectral characterizations of distance-regular graphs
© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth. (C) 2016 Published by Elsevier Inc.Peer ReviewedPostprint (author's final draft
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