18,005 research outputs found
On the maximal energy tree with two maximum degree vertices
For a simple graph , the energy is defined as the sum of the
absolute values of all eigenvalues of its adjacent matrix. For
and , denote by (or simply ) the tree formed from
a path on vertices by attaching 's on each end of the
path , and (or simply ) the tree formed from
by attaching 's on an end of the and
's on the vertex next to the end. In [X. Li, X. Yao, J. Zhang
and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math.
Chem. 45(2009), 962--973], Li et al. proved that among trees of order with
two vertices of maximum degree , the maximal energy tree is either the
graph or the graph , where . However, they
could not determine which one of and is the maximal energy tree.
This is because the quasi-order method is invalid for comparing their energies.
In this paper, we use a new method to determine the maximal energy tree. It
turns out that things are more complicated. We prove that the maximal energy
tree is for and any , while the maximal energy
tree is for and any . Moreover, for , the
maximal energy tree is for all but , for which is
the maximal energy tree. For , the maximal energy tree is for
all but is odd and , for which is the
maximal energy tree. For , the maximal energy tree is for all
but , for which is the maximal energy tree. One can
see that for most , is the maximal energy tree, is a
turning point, and and 4 are exceptional cases.Comment: 16 page
Effect of charged impurities on graphene thermoelectric power near the Dirac point
In graphene devices with a varying degree of disorders as characterized by
their carrier mobility and minimum conductivity, we have studied the
thermoelectric power along with the electrical conductivity over a wide range
of temperatures. We have found that the Mott relation fails in the vicinity of
the Dirac point in high-mobility graphene. By properly taking account of the
high temperature effects, we have obtained good agreement between the Boltzmann
transport theory and our experimental data. In low-mobility graphene where the
charged impurities induce relatively high residual carrier density, the Mott
relation holds at all gate voltages
A Two-stage Polynomial Method for Spectrum Emissivity Modeling
Spectral emissivity is a key in the temperature measurement by radiation methods, but not easy to determine in a combustion environment, due to the interrelated influence of temperature and wave length of the radiation. In multi-wavelength radiation thermometry, knowing the spectral emissivity of the material is a prerequisite. However in many circumstances such a property is a complex function of temperature and wavelength and reliable models are yet to be sought. In this study, a two stages partition low order polynomial fitting is proposed for multi-wavelength radiation thermometry. In the first stage a spectral emissivity model is established as a function of temperature; in the second stage a mathematical model is established to describe the dependence of the coefficients corresponding to the wavelength of the radiation. The new model is tested against the spectral emissivity data of tungsten, and good agreement was found with a maximum error of 0.64
On a conjecture about tricyclic graphs with maximal energy
For a given simple graph , the energy of , denoted by , is defined as the sum of the absolute values of all eigenvalues of its
adjacency matrix, which was defined by I. Gutman. The problem on determining
the maximal energy tends to be complicated for a given class of graphs. There
are many approaches on the maximal energy of trees, unicyclic graphs and
bicyclic graphs, respectively. Let denote the graph with vertices obtained from three copies of and a path by
adding a single edge between each of two copies of to one endpoint of the
path and a single edge from the third to the other endpoint of the
. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P.
Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it
Europ. J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following
conjecture: Let be a tricyclic graphs on vertices with or
, then with equality
if and only if . Let denote the set of all
connected bipartite tricyclic graphs on vertices with three vertex-disjoint
cycles , and , where . In this paper, we try to
prove that the conjecture is true for graphs in the class ,
but as a consequence we can only show that this is true for most of the graphs
in the class except for 9 families of such graphs.Comment: 32 pages, 12 figure
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