1,007 research outputs found
An algebraic formulation of the graph reconstruction conjecture
The graph reconstruction conjecture asserts that every finite simple graph on
at least three vertices can be reconstructed up to isomorphism from its deck -
the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important
tool in graph reconstruction. Roughly speaking, given the deck of a graph
and any finite sequence of graphs, it gives a linear constraint that every
reconstruction of must satisfy.
Let be the number of distinct (mutually non-isomorphic) graphs on
vertices, and let be the number of distinct decks that can be
constructed from these graphs. Then the difference measures
how many graphs cannot be reconstructed from their decks. In particular, the
graph reconstruction conjecture is true for -vertex graphs if and only if
.
We give a framework based on Kocay's lemma to study this discrepancy. We
prove that if is a matrix of covering numbers of graphs by sequences of
graphs, then . In particular, all
-vertex graphs are reconstructible if one such matrix has rank . To
complement this result, we prove that it is possible to choose a family of
sequences of graphs such that the corresponding matrix of covering numbers
satisfies .Comment: 12 pages, 2 figure
Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot
We consider a two-dimensional massless Dirac operator in the presence of
a perturbed homogeneous magnetic field and a scalar electric
potential . For , , and , , both decaying at infinity, we show that
states in the discrete spectrum of are superexponentially localized. We
establish the existence of such states between the zeroth and the first Landau
level assuming that V=0. In addition, under the condition that is
rotationally symmetric and that satisfies certain analyticity condition on
the angular variable, we show that states belonging to the discrete spectrum of
are Gaussian-like localized
On the convergence of eigenfunctions to threshold energy states
We prove the convergence in certain weighted spaces in momentum space of
eigenfunctions of H = T-lambda*V as the energy goes to an energy threshold. We
do this for three choices of kinetic energy T, namely the non-relativistic
Schr"odinger operator, the pseudorelativistc operator sqrt{-\Delta+m^2}-m, and
the Dirac operator.Comment: 15 pages; references and comments added (e.g., Remark 3
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