On the maximal energy tree with two maximum degree vertices

For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix. For $\Delta\geq 3$ and $t\geq 3$, denote by $T_a(\Delta,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices by attaching $\Delta-1$ $P_2$'s on each end of the path $P_t$, and $T_b(\Delta, t)$ (or simply $T_b$) the tree formed from $P_{t+2}$ by attaching $\Delta-1$ $P_2$'s on an end of the $P_{t+2}$ and $\Delta -2$ $P_2$'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 $n$ with two vertices of maximum degree $\Delta$, the maximal energy tree is either the graph $T_a$ or the graph $T_b$, where $t=n+4-4\Delta\geq 3$. However, they could not determine which one of $T_a$ and $T_b$ 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 $T_b$ for $\Delta\geq 7$ and any $t\geq 3$, while the maximal energy tree is $T_a$ for $\Delta=3$ and any $t\geq 3$. Moreover, for $\Delta=4$, the maximal energy tree is $T_a$ for all $t\geq 3$ but $t=4$, for which $T_b$ is the maximal energy tree. For $\Delta=5$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t$ is odd and $3\leq t\leq 89$, for which $T_a$ is the maximal energy tree. For $\Delta=6$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t=3,5,7$, for which $T_a$ is the maximal energy tree. One can see that for most $\Delta$, $T_b$ is the maximal energy tree, $\Delta=5$ is a turning point, and $\Delta=3$ 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