707 research outputs found
Fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees
AbstractA bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every pair of vertices that are in different parts of the graph. It is well known that Cay(Sn,B) is Hamiltonian laceable, where Sn is the symmetric group on {1,2,…,n} and B is a generating set consisting of transpositions of Sn. In this paper, we show that for any F⊆E(Cay(Sn,B)), if |F|≤n−3 and n≥4, then there exists a Hamiltonian path in Cay(Sn,B)−F joining every pair of vertices that are in different parts of the graph. The result is optimal with respect to the number of edge faults
Magnetic field effects in energy relaxation mediated by Kondo impurities
We study the energy distribution function of quasiparticles in voltage biased
mesoscopic wires in presence of magnetic impurities and applied magnetic field.
The system is described by a Boltzmann equation where the collision integral is
determined by coupling to spin 1/2 impurities. We derive an effective coupling
to a dissipative spin system which is valid well above Kondo temperature in
equilibrium or for sufficiently smeared distribution functions in
non-equilibrium. For low magnetic field an enhancement of energy relaxation is
found whereas for larger magnetic fields the energy relaxation decreases again
meeting qualitatively the experimental findings by Anthore et al.
(cond-mat/0109297). This gives a strong indication that magnetic impurities are
in fact responsible for the enhanced energy relaxation in copper wires. The
quantitative comparison, however, shows strong deviations for energy relaxation
with small energy transfer whereas the large energy transfer regime is in
agreement with our findings.Comment: 14 pages, 8 figure
Combinatorial solutions to the Hamiltonian constraint in (2+1)-dimensional Ashtekar gravity
Dirac's quantization of the (2+1)-dimensional analog of Ashtekar's approach
to quantum gravity is investigated. After providing a diffeomorphism-invariant
regularization of the Hamiltonian constraint, we find a set of solutions to
this Hamiltonian constraint which is a generalization of the solution
discovered by Jacobson and Smolin. These solutions are given by particular
linear combinations of the spin network states. While the classical
counterparts of these solutions have degenerate metric, due to a \lq quantum
effect' the area operator has nonvanishing action on these states. We also
discuss how to extend our results to (3+1)-dimensions.Comment: 41 pages Latex (2 figures available as a postscript file
Non ambiguous structures on 3-manifolds and quantum symmetry defects
The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic
oriented cusped -manifolds can be split in a "symmetrization" factor and a
"reduced" state sum. We show that these factors are invariants on their own,
that we call "symmetry defects" and "reduced QHI", provided the manifolds are
endowed with an additional "non ambiguous structure", a new type of
combinatorial structure that we introduce in this paper. A suitably normalized
version of the symmetry defects applies to compact -manifolds endowed with
-characters, beyond the case of cusped manifolds. Given a
manifold with non empty boundary, we provide a partial "holographic"
description of the non-ambiguous structures in terms of the intrinsic geometric
topology of . Special instances of non ambiguous structures can be
defined by means of taut triangulations, and the symmetry defects have a
particularly nice behaviour on such "taut structures". Natural examples of taut
structures are carried by any mapping torus with punctured fibre of negative
Euler characteristic, or by sutured manifold hierarchies. For a cusped
hyperbolic -manifold which fibres over , we address the question of
determining whether the fibrations over a same fibered face of the Thurston
ball define the same taut structure. We describe a few examples in detail. In
particular, they show that the symmetry defects or the reduced QHI can
distinguish taut structures associated to different fibrations of . To
support the guess that all this is an instance of a general behaviour of state
sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we
describe similar results about reduced Turaev-Viro invariants.Comment: 58 pages, 32 figures; exposition improved, ready for publicatio
Structure of Topological Lattice Field Theories in Three Dimensions
We construct and classify topological lattice field theories in three
dimensions. After defining a general class of local lattice field theories, we
impose invariance under arbitrary topology-preserving deformations of the
underlying lattice, which are generated by two new local lattice moves.
Invariant solutions are in one--to--one correspondence with Hopf algebras
satisfying a certain constraint. As an example, we study in detail the
topological lattice field theory corresponding to the Hopf algebra based on the
group ring \C[G], and show that it is equivalent to lattice gauge theory at
zero coupling, and to the Ponzano--Regge theory for SU(2).Comment: 63 pages, 46 figure
Constructing disjoint Steiner trees in Sierpi\'{n}ski graphs
Let be a graph and with . Then the trees
in are \emph{internally disjoint Steiner trees}
connecting (or -Steiner trees) if and
for every pair of distinct integers , . Similarly, if we only have the condition but without the condition , then they are
\emph{edge-disjoint Steiner trees}. The \emph{generalized -connectivity},
denoted by , of a graph , is defined as
,
where is the maximum number of internally disjoint -Steiner
trees. The \emph{generalized local edge-connectivity} is the
maximum number of edge-disjoint Steiner trees connecting in . The {\it
generalized -edge-connectivity} of is defined as
. These
measures are generalizations of the concepts of connectivity and
edge-connectivity, and they and can be used as measures of vulnerability of
networks. It is, in general, difficult to compute these generalized
connectivities. However, there are precise results for some special classes of
graphs. In this paper, we obtain the exact value of
for , and the exact value of for
, where is the Sierpi\'{n}ski graphs with order
. As a direct consequence, these graphs provide additional interesting
examples when . We also study the
some network properties of Sierpi\'{n}ski graphs
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