2,855 research outputs found
On Weakly Distinguishing Graph Polynomials
A univariate graph polynomial P(G;X) is weakly distinguishing if for almost
all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that
the clique polynomial and the independence polynomial are weakly
distinguishing. Furthermore, we show that generating functions of induced
subgraphs with property C are weakly distinguishing provided that C is of
bounded degeneracy or tree-width. The same holds for the harmonious chromatic
polynomial
Proper caterpillars are distinguished by their symmetric chromatic function
This paper deals with the so-called Stanley conjecture, which asks whether
they are non-isomorphic trees with the same symmetric function generalization
of the chromatic polynomial. By establishing a correspondence between
caterpillars trees and integer compositions, we prove that caterpillars in a
large class (we call trees in this class proper) have the same symmetric
chromatic function generalization of the chromatic polynomial if and only if
they are isomorphic
Crystal structure on rigged configurations
Rigged configurations are combinatorial objects originating from the Bethe
Ansatz, that label highest weight crystal elements. In this paper a new
unrestricted set of rigged configurations is introduced for types ADE by
constructing a crystal structure on the set of rigged configurations. In type A
an explicit characterization of unrestricted rigged configurations is provided
which leads to a new fermionic formula for unrestricted Kostka polynomials or
q-supernomial coefficients. The affine crystal structure for type A is obtained
as well.Comment: 20 pages, 1 figure, axodraw and youngtab style file necessar
On distinguishing trees by their chromatic symmetric functions
Let be an unrooted tree. The \emph{chromatic symmetric function} ,
introduced by Stanley, is a sum of monomial symmetric functions corresponding
to proper colorings of . The \emph{subtree polynomial} , first
considered under a different name by Chaudhary and Gordon, is the bivariate
generating function for subtrees of by their numbers of edges and leaves.
We prove that , where is the Hall inner
product on symmetric functions and is a certain symmetric function that
does not depend on . Thus the chromatic symmetric function is a stronger
isomorphism invariant than the subtree polynomial. As a corollary, the path and
degree sequences of a tree can be obtained from its chromatic symmetric
function. As another application, we exhibit two infinite families of trees
(\emph{spiders} and some \emph{caterpillars}), and one family of unicyclic
graphs (\emph{squids}) whose members are determined completely by their
chromatic symmetric functions.Comment: 16 pages, 3 figures. Added references [2], [13], and [15
Computational Hardness of Certifying Bounds on Constrained PCA Problems
Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics.
Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896
Quantum Spectrum Testing
In this work, we study the problem of testing properties of the spectrum of a
mixed quantum state. Here one is given copies of a mixed state
and the goal is to distinguish whether 's
spectrum satisfies some property or is at least -far in
-distance from satisfying . This problem was promoted in
the survey of Montanaro and de Wolf under the name of testing unitarily
invariant properties of mixed states. It is the natural quantum analogue of the
classical problem of testing symmetric properties of probability distributions.
Here, the hope is for algorithms with subquadratic copy complexity in the
dimension . This is because the "empirical Young diagram (EYD) algorithm"
can estimate the spectrum of a mixed state up to -accuracy using only
copies. In this work, we show that given a
mixed state : (i) copies
are necessary and sufficient to test whether is the maximally mixed
state, i.e., has spectrum ; (ii)
copies are necessary and sufficient to test with
one-sided error whether has rank , i.e., has at most nonzero
eigenvalues; (iii) copies are necessary and
sufficient to distinguish whether is maximally mixed on an
-dimensional or an -dimensional subspace; and (iv) The EYD
algorithm requires copies to estimate the spectrum of
up to -accuracy, nearly matching the known upper bound. In
addition, we simplify part of the proof of the upper bound. Our techniques
involve the asymptotic representation theory of the symmetric group; in
particular Kerov's algebra of polynomial functions on Young diagrams.Comment: 70 pages, 6 figure
Derived equivalence classification of the cluster-tilted algebras of Dynkin type E
We obtain a complete derived equivalence classification of the cluster-tilted
algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and
E8 which turn out to fall into 6, 14, 15 derived equivalence classes,
respectively. This classification can be achieved computationally and we
outline an algorithm which has been implemented to carry out this task. We also
make the classification explicit by giving standard forms for each derived
equivalence class as well as complete lists of the algebras contained in each
class; as these lists are quite long they are provided as supplementary
material to this paper. From a structural point of view the remarkable outcome
of our classification is that two cluster-tilted algebras of Dynkin type E are
derived equivalent if and only if their Cartan matrices represent equivalent
bilinear forms over the integers which in turn happens if and only if the two
algebras are connected by a sequence of "good" mutations. This is reminiscent
of the derived equivalence classification of cluster-tilted algebras of Dynkin
type A, but quite different from the situation in Dynkin type D where a
far-reaching classification has been obtained using similar methods as in the
present paper but some very subtle questions are still open.Comment: 19 pages. v4: completely rewritten version, to appear in Algebr.
Represent. Theory. v3: Main theorem strengthened by including "good"
mutations (cf. also arXiv:1001.4765). Minor editorial changes. v2: Third
author added. Major revision. All questions left open in the earlier version
by the first two authors are now settled in v2 and the derived equivalence
classification is completed. arXiv admin note: some text overlap with
arXiv:1012.466
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