The Minimum Spectral Radius of Graphs with the Independence Number

In this paper, we investigate some properties of the Perron vector of connected graphs. These results are used to characterize that all extremal connected graphs with having the minimum (maximum) spectra radius among all connected graphs of order $n=k\alpha$ with the independence number $\alpha$, respectively.Comment: 28 pages, 3 figure

The Wiener and Terminal Wiener indices of trees

Heydari \cite{heydari2013} presented very nice formulae for the Wiener and terminal Wiener indices of generalized Bethe trees. It is pity that there are some errors for the formulae. In this paper, we correct these errors and characterize all trees with the minimum terminal Wiener index among all the trees of order $n$ and with maximum degree $\Delta$.Comment: 13 page

Branching ratios, CPCP asymmetries and polarizations of B→ψ(2S)VB\rightarrow \psi(2S) V decays

We analyzed the nonleptonic decays $B/B_s\to \psi(2S) V$ with $V=(\rho, \omega, K^{*}, \phi)$ by employing the perturbative QCD (PQCD) factorization approach. Here the branching ratios, the $CP$ asymmetries and the complete set of polarization observables are investigated systematically. Besides the traditional contributions from the factorizable and nonfactorizable diagrams at the leading order, the next-to-leading order (NLO) vertex corrections could also provide considerable contributions. The PQCD predictions for the branching ratios of the $B_{(s)}\to \psi(2S)K^{*}, \psi(2S) \phi$ decays are consistent with the measured values within errors. As for $B\to \psi(2S) \rho, \psi(2S) \omega$ decays, the branching ratios can reach the order of $10^{-5}$ and could be measured in the LHCb and Belle-II experiments. The numerical results show that the direct $CP$ asymmetries of the considered decays are very small. Thus the observation of any large direct $CP$ asymmetry for these decays will be a signal for new physics. The mixing induced $CP$ asymmetries in the neutral modes are very close to $\sin 2\beta_{(s)}$, which suggests that these channels can give a cross-check on the measurement of the Cabbibo-Kobayashi-Maskawa (CKM) angle $\beta$ and $\beta_s$. We found that the longitudinal polarization fractions $f_0$ are suppressed to $\sim 50\%$ due to the large nonfactorizable contributions. The magnitudes and phases of the two transverse amplitudes $\mathcal {A}_{\parallel}$ and $\mathcal {A}_{\perp}$ are roughly equal, which is an indication for the approximate light quark helicity conservation in these decays. The overall polarization observables of $B\to \psi(2S) K^{*0}$ and $B_s\to \psi(2S) \phi$ channels are also in good agreement with the experimental measurements as reported by LHCb and BaBar. Other results can also be tested by the LHCb and Belle-II experiments.Comment: 14 pages, 1 figure, 6 table

Bulk Viscosity of dual Fluid at Finite Cutoff Surface via Gravity/Fluid correspondence in Einstein-Maxwell Gravity

Based on the previous paper arXiv:1207.5309, we investigate the possibility to find out the bulk viscosity of dual fluid at the finite cutoff surface via gravity/fluid correspondence in Einstein-Maxwell gravity. We find that if we adopt new conditions to fix the undetermined parameters contained in the stress tensor and charged current of the dual fluid, two new terms appear in the stress tensor of the dual fluid. One new term is related to the bulk viscosity term, while the other can be related to the perturbation of energy density. In addition, since the parameters contained in the charged current are the same, the charged current is not changed.Comment: 15 pages, no figure, typos corrected, new references and comments added, version accepted by PL