On asymptotic behavior of Bell polynomials and concentration of vertex degree of large random graphs

We study concentration properties of vertex degrees of $n$-dimensional Erdos-R'enyi random graphs with the edge probability $\rho \mathrm{/}n$ by means of high moments of these random variables in the limit when $n$ and $\rho$ tend to infinity. These moments are asymptotically close to one-variable Bell polynomials ${\mathcal{B}}_k(\rho ),k\in \mathbf{N}$ that represent moments of the Poisson probability distribution $\mathcal{P}(\rho )$. We study asymptotic behavior of the Bell polynomials and modified Bell polynomials for large values of $k$ and $\rho$ with the help of the local limit theorem for auxiliary random variables. Using the results obtained, we get the upper bounds for the deviation probabilities of the normalized maximal vertex degree of the Erdos-R'enyi random graphs in the limit $n,\rho \to \mathrm{\infty }$ such that the ratio $\rho \mathrm{/}\log n$ remains finite or infinitely increases.