# 图论|MATH0029 Graph Theory代写

0

The course aims to introduce students to discrete mathematics, a fundamental part of mathematics with many applications in computer science and related areas. The course provides an introduction to graph theory and combinatorics, the two cornerstones of discrete mathematics. The course will be offered to third or fourth year students taking Mathematics degrees, and might also be suitable for students from other departments. There will be an emphasis on extremal results and a variety of methods.

$$(1-\epsilon) p q n=(1-\epsilon) c n \leq \sum_{i=1}^{r}\left|X_{i}\right|=r \ell \leq q \ell$$
whence $(1-\epsilon) n \leq \epsilon \ell$ and hence $j<n \leq 2 \epsilon \ell$. Because $\left|Y_{s}\right| \geq 4^{-d_{j, k} \ell} \geq 4^{-\Delta} \ell$ and $\left(\Delta^{2}+2\right) \epsilon<4^{-\Delta}$,

$$\left|Y_{s}\right|-\Delta^{2} \epsilon \ell-j>4^{-\Delta} \ell-\Delta^{2} \epsilon \ell-2 \epsilon \ell=\left(4^{-\Delta}-\left(\Delta^{2}+2\right) \epsilon\right) \ell>0$$
Also, because $\epsilon<\frac{1}{4}$ and $\left|Y_{t}\right| \geq 4^{-d_{j, k} \ell}$,
$$\left(\frac{1}{2}-\epsilon\right)\left|Y_{t}\right| \geq \frac{1}{4}\left(4^{-d_{j, k}} \ell\right)=4^{-d_{j, k}-1} \ell=4^{-d_{j+1, k} \ell}$$

## MATH0029 COURSE NOTES ：

For a pair ${X, Y}$ of disjoint sets of vertices of a graph $G$, we define its index of regularity by:
$$\rho(X, Y):=|X | Y|(d(X, Y))^{2}$$
This index is nonnegative. We extend it to a family $\mathcal{P}$ of disjoint subsets of $V$ by setting:
$$\rho(\mathcal{P}):=\sum_{X, Y \in \mathcal{P}} \rho(X, Y)$$
In the case where $\mathcal{P}$ is a partition of $V$, we have:
$$\rho(\mathcal{P})=\sum_{\substack{X \in Y \in \mathcal{P} \ X \neq Y}}|X | Y|(d(X, Y))^{2} \leq \sum_{\substack{X, Y \in \mathcal{P} \ X \neq Y}}|X||Y|<\frac{n^{2}}{2}$$

# 图论作业代写Graph Theory代考

0

## 代写图论作业代写Graph Theory

### 重图Multigraph代写

• 计算机科学Computer science
• 计算神经科学Computational neuroscience

## 图论的历史

The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler’s formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L’Huilier,and represents the beginning of the branch of mathematics known as topology.

## 图论课后作业代写

Proof It clearly suffices to prove the corollary for connected graphs. Let $G$ be a simple connected planar graph with $n \geq 3$. Consider any planar embedding $\widetilde{G}$ of $G$. Because $G$ is simple and connected, on at least three vertices, $d(f) \geq 3$ for all $f \in F(\widetilde{G})$. Therefore, by Theorem $10.10$ and Euler’s Formula (10.2)
$$2 m=\sum_{f \in F(\widetilde{G})} d(f) \geq 3 f(\widetilde{G})=3(m-n+2)$$
or, equivalently,
$$m \leq 3 n-6$$
Equality holds in (10.4) if and only if it holds in (10.3), that is, if and only if $d(f)=3$ for each $f \in F(\widetilde{G})$.