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

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