3,386 research outputs found
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
Left-Right Entanglement Entropy of Dp-branes
We compute the left-right entanglement entropy for Dp-branes in string
theory. We employ the CFT approach to string theory Dp-branes, in particular,
its presentation as coherent states of the closed string sector. The
entanglement entropy is computed as the von Neumann entropy for a density
matrix resulting from integration over the left-moving degrees of freedom. We
discuss various crucial ambiguities related to sums over spin structures and
argue that different choices capture different physics; however, we advance a
themodynamic argument that seems to favor a particular choice of replica. We
also consider Dp branes on compact dimensions and verify that the effects of
T-duality act covariantly on the Dp brane entanglement entropy. We find that
generically the left-right entanglement entropy provides a suitable
generalization of boundary entropy and of the D-brane tension.Comment: 20 pages, 3 figures. v2: A thermodynamic argument favoring a
particular treatment of spin structures is advanced; one figure improved and
references adde
Left-Right Entanglement Entropy of Boundary States
We study entanglement entropy of boundary states in a free bosonic conformal
field theory. A boundary state can be thought of as composed of a particular
combination of left and right-moving modes of the two-dimensional conformal
field theory. We investigate the reduced density matrix obtained by tracing
over the right-moving modes in various boundary states. We consider Dirichlet
and Neumann boundary states of a free noncompact as well as a compact boson.
The results for the entanglement entropy indicate that the reduced system can
be viewed as a thermal CFT gas. Our findings are in agreement and generalize
results in quantum mechanics and quantum field theory where coherent states can
also be considered. In the compact case we verify that the entanglement entropy
expressions are consistent with T-duality.Comment: 15 pages, no figures. v2 References added, typos fixe
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