A maximum-degree tail with an explicit logarithmic exponent

Since $\mu =np>0$ and $0<p<1$, $n\ge 1$, so the vertex set $[n]$ is nonempty. Vertex degree counts incident edges and $d_{\max }$ is the maximum of those degrees (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree). Thus a vertex degree is the sum of its $n-1$ independent Bernoulli edge indicators (The Erdős-Rényi finite random graph G(n,p)). Each Bernoulli indicator is $1$ with probability $p$ and $0$ with probability $1-p$ (Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials).

Finite product weights factor over independent edge coordinates (The finite product of finite probability spaces).

For outcome weights $w(\omega )\ge 0$, expectation is $\mathbb{E}X={\sum }_{\omega \in \mathrm{\Omega }}X(\omega )w(\omega )$ (Expectation of a real random variable on a finite probability space), while an event $E\subseteq \mathrm{\Omega }$ has probability $\mathbb{P}(E)={\sum }_{\omega \in E}w(\omega )$ (Finite probability spaces, outcome weights, events, and event probabilities).

The exponential has series $\exp (x)={\sum }_{j\ge 0}{x}^j\mathrm{/}j!$, and $e=\exp (1)$ (The real exponential function and the number e by a power series).

The exponential satisfies its addition formula (The exponential addition formula exp(x+y)=exp(x)exp(y)).

The bound $1+x\le \exp (x)$ holds for every real $x$ (1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp)).

Real powers obey $n^a=\exp (a\log n)$ (Real powers for positive bases, with the zero-base positive-exponent convention).

For $x>0$, $\log x$ is the unique real number with $\exp (\log x)=x$ (The natural logarithm as the inverse of the exponential function).

The exponential is strictly increasing on the real line (The exponential function is strictly increasing).

For real $r$ with $\mathrm{\mid }r\mathrm{\mid }<1$, the geometric series satisfies ${\sum }_{k=0}^{\mathrm{\infty }}{r}^k=1\mathrm{/}(1-r)$. At $r=1\mathrm{/}2$ this gives ${\sum }_{k=1}^{\mathrm{\infty }}{2}^{-k}=1$ (For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges).