We consider the stochastic equation \[ X_t=x_0+\int _0^t b(u,X_{u})\mathrm{d}B_u,\quad t\ge 0, \] where $B$ is a one-dimensional Brownian motion, $x_0\in \mathbb{R}$ is the initial value, and $b\:[0,\infty )\times \mathbb{R}\rightarrow \mathbb{R}$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$, beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution.
Let µ_{n-1}(G) be the algebraic connectivity, and let µ_{1}(G) be the Laplacian spectral radius of a k-connected graph G with n vertices and m edges. In this paper, we prove that {\mu _{n - 1}}(G) \geqslant \frac{{2n{k^2}}}{{(n(n - 1) - 2m)(n + k - 2) + 2{k^2}}} , with equality if and only if G is the complete graph Kn or Kn − e. Moreover, if G is non-regular, then {\mu _1}(G) < 2\Delta - \frac{{2(n\Delta - 2m){k^2}}}{{2(n\Delta - 2m)({n^2} - 2n + 2k) + n{k^2}}} , where ▵ stands for the maximum degree of G. Remark that in some cases, these two inequalities improve some previously known results., Xiaodan Chen, Yaoping Hou., and Obsahuje seznam literatury