3,118 research outputs found
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields
Let be a smooth vector field on the plane, that is a map from the plane
to the unit circle. We study sufficient conditions for the boundedness of the
Hilbert transform
\operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon
f(x-yv(x)) \frac{dy}y where is a suitably chosen parameter,
determined by the smoothness properties of the vector field. It is a
conjecture, due to E.\thinspace M.\thinspace Stein, that if is Lipschitz,
there is a positive for which the transform above is bounded on . Our principal result gives a sufficient condition in terms of the
boundedness of a maximal function associated to . This sufficient condition
is that this new maximal function be bounded on some , for some . We show that the maximal function is bounded from to weak for all Lipschitz maximal function. The relationship between our results
and other known sufficient conditions is explored.Comment: 92 pages, 20+ figures. Final version of the paper. To appear in
Memoirs AM
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