6,152 research outputs found
Graphs with small diameter determined by their -spectra
Let be a connected graph with vertex set
. The distance matrix is the matrix indexed by the vertices of where denotes the
distance between the vertices and . Suppose that
are the distance
spectrum of . The graph is said to be determined by its -spectrum if
with respect to the distance matrix , any graph having the same spectrum
as is isomorphic to . In this paper, we give the distance characteristic
polynomial of some graphs with small diameter, and also prove that these graphs
are determined by their -spectra
Constraints on the Asymptotic Baryon Fractions of Galaxy Clusters at Large Radii
While X-ray measurements have so far revealed an increase in the
volume-averaged baryon fractions of galaxy clusters with cluster radii
, should asymptotically reach a universal value ,
provided that clusters are representative of the Universe. In the framework of
hydrostatic equilibrium for intracluster gas, we have derived the necessary
conditions for : The X-ray surface brightness profile
described by the model and the temperature profile approximated by the
polytropic model should satisfy and
for , respectively, which sets
a stringent limit to the polytropic index: . In particular, a
mildly increasing temperature with radius is required if the observationally
fitted parameter is in the range . It is likely that a
reliable determination of the universal baryon fraction can be achieved in the
small clusters because the disagreement between the exact and
asymptotic baryon fractions for clusters with breaks down at rather
large radii (\ga30r_c) where hydrostatic equilibrium has probably become
inapplicable. We further explore how to obtain the asymptotic value
of baryon fraction from the X-ray measurement made primarily over
the finite central region of a cluster. We demonstrate our method using a
sample of 19 strong lensing clusters, which enables us to place a useful
constraint on : .
An optimal estimate of based on three cooling flow clusters with
or .Comment: 6 pages + 4 figures; accepted for publication in MNRA
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