8,025 research outputs found
Distances on the tropical line determined by two points
Let . Write if is a multiple of
. Two different points and in uniquely
determine a tropical line , passing through them, and stable under
small perturbations. This line is a balanced unrooted semi--labeled tree on
leaves. It is also a metric graph.
If some representatives and of and are the first and second
columns of some real normal idempotent order matrix , we prove that the
tree is described by a matrix , easily obtained from . We also
prove that is caterpillar. We prove that every vertex in
belongs to the tropical linear segment joining and . A vertex, denoted
, closest (w.r.t tropical distance) to exists in . Same for
. The distances between pairs of adjacent vertices in and the
distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the
matrix . In addition, if and are generic, then the tree
is trivalent. The entries of are differences (i.e., sum of principal
diagonal minus sum of secondary diagonal) of order 2 minors of the first two
columns of .Comment: New corrected version. 31 pages and 9 figures. The main result is
theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7
was obtained with A. Jim\'enez; see Arxiv 1205.416
Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width
We prove that every graph of rank-width is a pivot-minor of a graph of
tree-width at most . We also prove that graphs of rank-width at most 1,
equivalently distance-hereditary graphs, are exactly vertex-minors of trees,
and graphs of linear rank-width at most 1 are precisely vertex-minors of paths.
In addition, we show that bipartite graphs of rank-width at most 1 are exactly
pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are
precisely pivot-minors of paths.Comment: 16 pages, 7 figure
Rearranging trees for robust consensus
In this paper, we use the H2 norm associated with a communication graph to
characterize the robustness of consensus to noise. In particular, we restrict
our attention to trees and by systematic attention to the effect of local
changes in topology, we derive a partial ordering for undirected trees
according to the H2 norm. Our approach for undirected trees provides a
constructive method for deriving an ordering for directed trees. Further, our
approach suggests a decentralized manner in which trees can be rearranged in
order to improve their robustness.Comment: Submitted to CDC 201
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