Concept of the locating-chromatic number of graphs is introduced firstly by Chartrand et al. in 2002. In this concept only limited for connected graphs. In 2004, Welyyanti et al. generalized the ...concept and notion of the locating-chromatic number of a graph such that it can be applied for a disconnected graph. In this paper, we determine the locating-chromatic number for disconnected graphs with path, stars, cycle, or double stars as it's components.
Given a properly edge-colored graph F, we call it rainbow if all of its edges receive different colors. The maximum number of edges, denoted by ex∗(n,F), in a properly edge-colored graph on n ...vertices which contains no rainbow F is called the rainbow Turán number of F. When the host graph is planar, we denote it as exP∗(n,F). A graph obtained by taking an edge with s vertices joining one of its end vertices and k vertices joining the other end vertex is called an (s,k)-double star, dented by Ss,k. In this paper, we study exP∗(n,S1,k) for all k except k=5 and exP∗(n,S2,2).
The merger of two massive (about 30 solar masses) black holes has been detected in gravitational waves. This discovery validates recent predictions that massive binary black holes would constitute ...the first detection. Previous calculations, however, have not sampled the relevant binary-black-hole progenitors--massive, low-metallicity binary stars--with sufficient accuracy nor included sufficiently realistic physics to enable robust predictions to better than several orders of magnitude. Here we report high-precision numerical simulations of the formation of binary black holes via the evolution of isolated binary stars, providing a framework within which to interpret the first gravitational-wave source, GW150914, and to predict the properties of subsequent binary-black-hole gravitational-wave events. Our models imply that these events form in an environment in which the metallicity is less than ten per cent of solar metallicity, and involve stars with initial masses of 40-100 solar masses that interact through mass transfer and a common-envelope phase. These progenitor stars probably formed either about 2 billion years or, with a smaller probability, 11 billion years after the Big Bang. Most binary black holes form without supernova explosions, and their spins are nearly unchanged since birth, but do not have to be parallel. The classical field formation of binary black holes we propose, with low natal kicks (the velocity of the black hole at birth) and restricted common-envelope evolution, produces approximately 40 times more binary-black-holes mergers than do dynamical formation channels involving globular clusters; our predicted detection rate of these mergers is comparable to that from homogeneous evolution channels. Our calculations predict detections of about 1,000 black-hole mergers per year with total masses of 20-80 solar masses once second-generation ground-based gravitational-wave observatories reach full sensitivity.
Abstract
Let is
H
a disconnected graph and
c
is the locating
k
-coloring of
H
. The locating-chromatic number of
H
, denoted by
X
L
'
(
H
), is the smallest
k
such that
H
admits a locating-coloring ...with
k
colors. In this paper, we study the locating-chromatic number of disconnected graph with path
P
4
and double stars
S
m,n
graph as its components.
We present fully general-relativistic simulations of binary neutron star mergers with a temperature and composition dependent nuclear equation of state. We study the dynamical mass ejection from both ...quasi-circular and dynamical-capture eccentric mergers. We systematically vary the level of our treatment of the microphysics to isolate the effects of neutrino cooling and heating and we compute the nucleosynthetic yields of the ejecta. We find that eccentric binaries can eject significantly more material than quasi-circular binaries and generate bright infrared and radio emission. In all our simulations the outflow is composed of a combination of tidally- and shock-driven ejecta, mostly distributed over a broad ∼60° angle from the orbital plane, and, to a lesser extent, by thermally driven winds at high latitudes. Ejecta from eccentric mergers are typically more neutron rich than those of quasi-circular mergers. We find neutrino cooling and heating to affect, quantitatively and qualitatively, composition, morphology, and total mass of the outflows. This is also reflected in the infrared and radio signatures of the binary. The final nucleosynthetic yields of the ejecta are robust and insensitive to input physics or merger type in the regions of the second and third r-process peaks. The yields for elements on the first peak vary between our simulations, but none of our models is able to explain the Solar abundances of first-peak elements without invoking additional first-peak contributions from either neutrino and viscously-driven winds operating on longer time-scales after the mergers, or from core-collapse supernovae.
The basic geometry of the Solar System-the shapes, spacings, and orientations of the planetary orbits-has long been a subject of fascination as well as inspiration for planet-formation theories. For ...exoplanetary systems, those same properties have only recently come into focus. Here we review our current knowledge of the occurrence of planets around other stars, their orbital distances and eccentricities, the orbital spacings and mutual inclinations in multiplanet systems, the orientation of the host star's rotation axis, and the properties of planets in binary-star systems.
We summarize our current knowledge of neutron-star masses and radii. Recent instrumentation and computational advances have resulted in a rapid increase in the discovery rate and precise timing of ...radio pulsars in binaries in the past few years, leading to a large number of mass measurements. These discoveries show that the neutron-star mass distribution is much wider than previously thought, with three known pulsars now firmly in the 1.9-2.0-M
mass range. For radii, large, high-quality data sets from X-ray satellites as well as significant progress in theoretical modeling led to considerable progress in the measurements, placing them in the 10-11.5-km range and shrinking their uncertainties, owing to a better understanding of the sources of systematic errors. The combination of the massive-neutron-star discoveries, the tighter radius measurements, and improved laboratory constraints of the properties of dense matter has already made a substantial impact on our understanding of the composition and bulk properties of cold nuclear matter at densities higher than that of the atomic nucleus, a major unsolved problem in modern physics.
Recent observations have shown that circumstellar and circumbinary discs in young stellar binaries are often misaligned with respect to the binary orbital plane. We analyse the tidal truncation of ...such misaligned discs due to torques applied to the disc at the Lindblad resonances from the tidal forcings of the binary. We consider eccentric binaries with arbitrary binary-disc inclination angles. We determine the dependence of the tidal forcing strengths on the binary parameters and show that they are complicated non-monotonic functions of eccentricity and inclination. We adopt a truncation criterion determined by the balance between resonant torque and viscous torque, and use it to calculate the outer radii of circumstellar discs and the inner radii of circumbinary discs. Misaligned circumstellar discs have systematically larger outer radii than aligned discs, and are likely to fill their Roche lobes if inclined by more than 45°–90°, depending on the binary mass ratio and disc viscosity parameter. Misaligned circumbinary discs generally have smaller inner radii than aligned discs, but the details depend sensitively on the binary and disc parameters.
White dwarfs are compact stars, similar in size to Earth but approximately 200,000 times more massive. Isolated white dwarfs emit most of their power from ultraviolet to near-infrared wavelengths, ...but when in close orbits with less dense stars, white dwarfs can strip material from their companions and the resulting mass transfer can generate atomic line and X-ray emission, as well as near- and mid-infrared radiation if the white dwarf is magnetic. However, even in binaries, white dwarfs are rarely detected at far-infrared or radio frequencies. Here we report the discovery of a white dwarf/cool star binary that emits from X-ray to radio wavelengths. The star, AR Scorpii (henceforth AR Sco), was classified in the early 1970s as a δ-Scuti star, a common variety of periodic variable star. Our observations reveal instead a 3.56-hour period close binary, pulsing in brightness on a period of 1.97 minutes. The pulses are so intense that AR Sco's optical flux can increase by a factor of four within 30 seconds, and they are also detectable at radio frequencies. They reflect the spin of a magnetic white dwarf, which we find to be slowing down on a 10
-year timescale. The spin-down power is an order of magnitude larger than that seen in electromagnetic radiation, which, together with an absence of obvious signs of accretion, suggests that AR Sco is primarily spin-powered. Although the pulsations are driven by the white dwarf's spin, they mainly originate from the cool star. AR Sco's broadband spectrum is characteristic of synchrotron radiation, requiring relativistic electrons. These must either originate from near the white dwarf or be generated in situ at the M star through direct interaction with the white dwarf's magnetosphere.
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