A semi-analytic calculation on the atmospheric tau neutrino flux in the GeV to TeV energy range

We present a semi-analytic calculation on the atmospheric tau neutrino flux in the GeV to TeV energy range. The atmospheric $\nu_{\tau}$ flux is calculated for the entire zenith angle range. This flux is contributed by the oscillations of muon neutrinos coming from the two-body $\pi, K$ decays and the three-body $\mu^{\pm}$ decays, and the intrinsic tau neutrino flux surviving the oscillations. The uncertainties in our calculations are discussed in detail. The implications of our result are also discussed.Comment: Revtex, 30 pages (including 13 figures); paper expanded and title slightly changed, to appear in Astroparticle Physic

Strongly Regular Graphs as Laplacian Extremal Graphs

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also provided.Comment: 11 pages, 4 figures, 1 tabl