1,053 research outputs found
On the extremal properties of the average eccentricity
The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity of a graph is the mean value
of eccentricities of all vertices of . The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease .
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
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
D-wave correlated Critical Bose Liquids in two dimensions
We develop a description of a new quantum liquid phase of interacting bosons
in 2d which possesses relative D-wave two-body correlations and which we call a
D-wave Bose Liquid (DBL). The DBL has no broken symmetries, supports gapless
boson excitations residing on "Bose surfaces" in momentum space, and exhibits
power law correlations with continuously variable exponents. While the DBL can
be constructed for bosons in the 2d continuum, the state only respects the
point group symmetries of the square lattice. On the lattice the DBL respects
all symmetries and does not require a particular filling. But lattice effects
allow a second distinct phase, a quasi-local variant which we call a D-wave
Local Bose Liquid (DLBL). Remarkably, the DLBL has short-range boson
correlations and hence no Bose surfaces, despite sharing gapless excitations
and other critical signatures with the DBL. Moreover, both phases are metals
with a resistance that vanishes as a power of the temperature. We establish
these results by constructing a class of many-particle wavefunctions for the
DBL, which are time reversal invariant analogs of Laughlin's quantum Hall
wavefunction for bosons at . A gauge theory formulation leads to a
simple mean field theory, and an N-flavor generalization enables incorporation
of gauge field fluctuations to deduce the properties of the DBL/DLBL; various
equal time correlation functions are in qualitative accord with the properties
inferred from the wavefunctions. We also identify a promising Hamiltonian which
might manifest the DBL or DLBL, and perform a variational study comparing to
other competing phases. We suggest how the DBL wavefunction can be generalized
to describe an itinerant non-Fermi liquid phase of electrons on the square
lattice with a no double occupancy constraint, a D-wave metal phase.Comment: 33 pages, 17 figure
Binary-black-hole initial data with nearly-extremal spins
There is a significant possibility that astrophysical black holes with
nearly-extremal spins exist. Numerical simulations of such systems require
suitable initial data. In this paper, we examine three methods of constructing
binary-black-hole initial data, focusing on their ability to generate black
holes with nearly-extremal spins: (i) Bowen-York initial data, including
standard puncture data (based on conformal flatness and Bowen-York extrinsic
curvature), (ii) standard quasi-equilibrium initial data (based on the
extended-conformal-thin-sandwich equations, conformal flatness, and maximal
slicing), and (iii) quasi-equilibrium data based on the superposition of
Kerr-Schild metrics. We find that the two conformally-flat methods (i) and (ii)
perform similarly, with spins up to about 0.99 obtainable at the initial time.
However, in an evolution, we expect the spin to quickly relax to a
significantly smaller value around 0.93 as the initial geometry relaxes. For
quasi-equilibrium superposed Kerr-Schild (SKS) data [method (iii)], we
construct initial data with \emph{initial} spins as large as 0.9997. We evolve
SKS data sets with spins of 0.93 and 0.97 and find that the spin drops by only
a few parts in 10^4 during the initial relaxation; therefore, we expect that
SKS initial data will allow evolutions of binary black holes with relaxed spins
above 0.99. [Abstract abbreviated; full abstract also mentions several
secondary results.
Massive disk formation in the tidal disruption of a neutron star by a nearly extremal black hole
Black hole-neutron star (BHNS) binaries are important sources of
gravitational waves for second-generation interferometers, and BHNS mergers are
also a proposed engine for short, hard gamma-ray bursts. The behavior of both
the spacetime (and thus the emitted gravitational waves) and the neutron star
matter in a BHNS merger depend strongly and nonlinearly on the black hole's
spin. While there is a significant possibility that astrophysical black holes
could have spins that are nearly extremal (i.e. near the theoretical maximum),
to date fully relativistic simulations of BHNS binaries have included
black-hole spins only up to =0.9, which corresponds to the black hole
having approximately half as much rotational energy as possible, given the
black hole's mass. In this paper, we present a new simulation of a BHNS binary
with a mass ratio and black-hole spin =0.97, the highest simulated
to date. We find that the black hole's large spin leads to the most massive
accretion disk and the largest tidal tail outflow of any fully relativistic
BHNS simulations to date, even exceeding the results implied by extrapolating
results from simulations with lower black-hole spin. The disk appears to be
remarkably stable. We also find that the high black-hole spin persists until
shortly before the time of merger; afterwards, both merger and accretion spin
down the black hole.Comment: 20 pages, 10 figures, submitted to Classical and Quantum Gravit
Extremal Properties of Complex Networks
We describe the structure of connected graphs with the minimum and maximum
average distance, radius, diameter, betweenness centrality, efficiency and
resistance distance, given their order and size. We find tight bounds on these
graph qualities for any arbitrary number of nodes and edges and analytically
derive the form and properties of such networks
Analytical and numerical study of the ground-track resonances of Dawn orbiting Vesta
The aim of Dawn mission is the acquisition of data from orbits around two
bodies, (4)Vesta and (1)Ceres, the two most massive asteroids. Due to the low
thrust propulsion, Dawn will slowly cross and transit through ground-track
resonances, where the perturbations on Dawn orbit may be significant. In this
context, to safety go the Dawn mission from the approach orbit to the lowest
science orbit, it is essential to know the properties of the crossed
resonances. This paper analytically investigates the properties of the major
ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters:
location of the equilibria, aperture of the resonances and period at the stable
equilibria. We develop a general method using an averaged Hamiltonian
formulation with a spherical harmonic approximation of the gravity field. If
the values of the gravity field coefficient change, our method stays correct
and applicable. We also discuss the effect of one uncertainty on the C20 and
C22 coefficients on the properties of the 1:1 resonance. These results are
checked by numerical tests. We determine that the increase of the eccentricity
appearing in the 2:3 resonance is due to the C22 and S22 coefficients.
Our method can be easily adapted to missions similar to Dawn because,
contrarily to the numerical results, the analytical formalism stays the same
and is valid for a wide range of physical parameters of the asteroid (namely
the shape and the mass) as well as for different spacecraft orbits.
Finally we numerically study the probability of the capture in resonance 1:1.
Our paper reproduces, explains and supplements the results of Tricarico and
Sykes (2010).Comment: 34 pages, 9 figures, 10 Table
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