# On some estimates for Erd\"os-R\`enyi random graph

## TL;DR

This paper derives asymptotic formulas for the factorial moments of the number of components in Erdős-Rényi graphs and provides bounds for the probabilities of connectivity and isolated vertices.

## Contribution

It introduces new asymptotic relations for factorial moments and bounds for key graph properties in Erdős-Rényi models using generating functions.

## Key findings

- Asymptotic relation for factorial moments of component counts
- Bounds for the probability of graph connectivity
- Bounds for the probability of isolated vertices

## Abstract

We consider a number $\nu_n$ of components in a random graph $G(n,p)$ with $n$ vertices, where the probability of an edge is equal to $p$. By operating with special generating functions we shows the next asymptotic relation for factorial moments of $\nu_n$: $$ \mathsf{E}(\nu_n-1)^{\underline s} = (1+o(1))\left( \frac 1p \sum\limits_{k=1}^\infty\frac{k^{k-2}}{k!}(npq^n)^k\right)^s + o(1) $$ as $n$ tends to $\infty$ and $q=1-p$. And the following inequations hold: $$ 1-2nq^{n-1} \le p_n\le\frac{1}{nq^n}, $$ $$ 1-\frac{1}{nq^n}\le pi_n\le nq^{n-1}, $$ where $p_n$ is the probability that $G(n,p)$ is connected and $pi_n$ is the probability that $G(n,p)$ has an isolated vertex.