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# 数学代写|随机图论代考Random Graph Theory代写|MAST30011 Extreme Characteristics

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## 数学代写|随机图论代考Random Graph Theory代写|Extreme Characteristics

In this section we will first discuss the threshold for $\mathbb{G}{n, p}$ to have diameter $d$, when $d \geq 2$ is a constant. The diameter of a connected graph $G$ is the maximum over distinct vertices $v, w$ of $\operatorname{dist}(v, w)$ where $\operatorname{dist}(v, w)$ is the minimum number of edges in a path from $v$ to $w$. The theorem below was proved independently by Burtin [172], [173] and by Bollobás [129]. The proof we give is due to Spencer [721]. Theorem 7.1. Let $d \geq 2$ be a fixed positive integer. Suppose that $c>0$ and $$p^d n^{d-1}=\log \left(n^2 / c\right) .$$ Then $$\lim {n \rightarrow \infty} \mathbb{P}\left(\operatorname{diam}\left(\mathbb{G}_{n, p}\right)=k\right)= \begin{cases}e^{-c / 2} & \text { if } k=d \ 1-e^{-c / 2} & \text { if } k=d+1 .\end{cases}$$
Proof. (a): w.h.p. $\operatorname{diam}(G) \geq d$.
Fix $v \in V$ and let
$$N_k(v)={w: \operatorname{dist}(v, w)=k} .$$

It follows from Theorem $3.4$ that w.h.p. for $0 \leq k<d$,
$$\left|N_k(v)\right| \leq \Delta^k \approx(n p)^k \approx(n \log n)^{k / d}=o(n) .$$
(b) w.h.p. $\operatorname{diam}(G) \leq d+1$
Fix $v, w \in[n]$. Then for $1 \leq k<d$, define the event
$$\mathscr{F}k=\left{\left|N_k(v)\right| \in I_k=\left[\left(\frac{n p}{2}\right)^k,(2 n p)^k\right]\right} .$$ Then for $k \leq\lceil d / 2\rceil$ we have \begin{aligned} & \mathbb{P}\left(\neg \mathscr{F}_k \mid \mathscr{F}_1, \ldots, \mathscr{F}{k-1}\right)= \ & =\mathbb{P}\left(\operatorname{Bin}\left(n-\sum_{i=0}^{k-1}\left|N_i(v)\right|, 1-(1-p)^{\left|N_{k-1}(v)\right|}\right) \notin I_k\right) \ & \leq \mathbb{P}\left(\operatorname{Bin}\left(n-o(n), \frac{3}{4}\left(\frac{n p}{2}\right)^{k-1} p\right) \leq\left(\frac{n p}{2}\right)^k\right) \ & \leq \quad, \quad+\mathbb{P}\left(\operatorname{Bin}\left(n-o(n), \frac{5}{4}(2 n p)^{k-1} p\right) \geq(2 n p)^k\right) \ & \leq \exp \left{-\Omega\left((n p)^k\right)\right} \ & =O\left(n^{-3}\right) . \end{aligned}
So with probability $1-O\left(n^{-3}\right)$,
$$\left|N_{\lfloor d / 2\rfloor}(v)\right| \geq\left(\frac{n p}{2}\right)^{\lfloor d / 2\rfloor} \text { and }\left|N_{\lceil d / 2\rceil}(w)\right| \geq\left(\frac{n p}{2}\right)^{\lceil d / 2\rceil}$$

## 数学代写|随机图论代考Random Graph Theory代写|Largest Independent Sets

Let $\alpha(G)$ denote the size of the largest independent set in a graph $G$.
Dense case
The following theorem was first proved by Matula [578].
Theorem 7.3. Suppose $0<p<1$ is a constant and $b=\frac{1}{1-p}$. Then w.h.p.
$$\alpha\left(\mathbb{G}_{n, p}\right) \approx 2 \log _b n .$$
Proof. Let $X_k$ be the number of independent sets of order $k$.
(i) Let
$$k=\left\lceil 2 \log _b n\right\rceil$$
Then,
\begin{aligned} \mathbb{E} X_k & =\left(\begin{array}{l} n \ k \end{array}\right)(1-p)^{\left(\begin{array}{l} k \ 2 \end{array}\right)} \ & \leq\left(\frac{n e}{k(1-p)^{1 / 2}}(1-p)^{k / 2}\right)^k \ & \leq\left(\frac{e}{k(1-p)^{1 / 2}}\right)^k \ & =o(1) . \end{aligned}
(ii) Let now
$$k=\left\lfloor 2 \log _b n-5 \log _b \log n\right\rfloor .$$

Let
$$\bar{\Delta}=\sum_{\substack{i, j \ S_i \sim S_j}} \mathbb{P}\left(S_i, S_j \text { are independent in } \mathbb{G}{n, p}\right),$$ where $S_1, S_2, \ldots, S{\left(\begin{array}{l}n \ k\end{array}\right.}$ are all the $k$-subsets of $[n]$ and $S_i \sim S_j$ iff $\left|S_i \cap S_j\right| \geq 2$. By Janson’s inequality, see Theorem 23.13,
$$\mathbb{P}\left(X_k=0\right) \leq \exp \left{-\frac{\left(\mathbb{E} X_k\right)^2}{2 \bar{\Delta}}\right} .$$
Here we apply the inequality in the context of $X_k$ being the number of $k$-cliques in the complement of $G_{n, p}$. The set $[N]$ will be the edges of the complete graph and the sets $D_i$ will the edges of the $k$-cliques. Now
\begin{aligned} & =\sum_{j=2}^k \frac{\left(\begin{array}{l} n-k \ k-j \end{array}\right)\left(\begin{array}{l} k \ j \end{array}\right)}{\left(\begin{array}{l} n \ k \end{array}\right)}(1-p)^{-\left(\begin{array}{l} j \ 2 \end{array}\right)} \ & =\sum_{j=2}^k u_j \ & \end{aligned}

## 数学代写|随机图论代考Random Graph Theory代写|Extreme Characteristics

$$p^d n^{d-1}=\log \left(n^2 / c\right)$$

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