# 数学代写|数学建模代写math modelling代考|Planar Graphs

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## 数学代写|数学建模代写math modelling代考|Planar Graphs

In the printing of TV and radio circuits, we want to ensure that the wires, all lying in a plane, should not intersect. In the graph of Figure 7.27(a) wires appear to intersect, but we can find an isomorphic graph in Figure 7.27(b) in which edges do not intersect. A graph which is such that we can draw a graph isomorphic to it in which edges do not intersect is called a planar graph.

A complete graph with five vertices is not planar (Figure 7.28a). We can draw nine of the edges so that these do not intersect (Figure 7.28b) but however we may draw, we cannot draw all ten edges without at least two of them intersecting. The proof of this depends on Jordan’s theorem that every simple closed curve divides the plane into two regions, one inside the curve and one outside the curve. $A B C D E$ in Figure $7.28(b)$ is a closed Jordan curve and we cannot draw three edges either inside it or outside it without intersecting.

A polygonal graph with $n$ vertices and $n$ straight or curved edges has $n$ vertices, $n$ edges, and two faces (one inside and one outside) so that for this graph
$$V-E+F=2$$
If we add on one edge another polygonal region of $r$ vertices, we increase the number of vertices by $r-2$, the number of edges by $r-1$, and the number of faces by 1 , so that the net increases in $V-E+F$ is zero and Eqn. (26) remains valid. It can be shown by using the principle of induction that (26) is valid for any polygonal graph with any number of regions.

To draw the dual graph $G^$ of $G$, we take a point inside each region and draw an edge through it intersecting one of the edges of the region. It is obvious that for this dual graph the number of vertices, edges, and faces is given by $$V^=F, E=E^, F^=V$$
so that
$$V^-E^+F^*=F-E+V=2$$
as expected.

## 数学代写|数学建模代写math modelling代考|Regular Solids

A polygonal graph $G$ is said to be completely regular if both $G$ and its dual $G^$ are regular, i.e., if the degree of each vertex of $G$ is the same (say $\rho$ ) and the degree of each vertex of $G^$ is the same (say $\left.\rho^\right)$. From this definition, it follows $$2 E=\rho V=\rho^ F$$

or
$$E=\frac{1}{2} \rho V, F=\frac{\rho}{\rho^*} V$$
Substituting Eqn. (30) in Eqn. (26)
$$V-\frac{1}{2} \rho V+\frac{\rho}{\rho *} V=2$$
or
$$V\left(2 \rho+2 \rho^-\rho \rho^\right)=4 \rho^*$$
Since $V, \rho, \rho^$ are positive integers $$2 \rho+2 \rho^-\rho \rho^*>0 \text { or }(\rho-2)(\rho *-2)<4$$ If $\rho>2, \rho^>2$, the only solutions of the inequality Eqn. (33) are $\rho=3, \rho^=3 ; \rho=3$; $\rho^=4 ; \rho=3 ; \rho^=5 ; \rho=4, \rho^=3 ; \rho=5, \rho^=3$. Substituting in Eqns. (32) and (30), we get the table and graphs
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline & $\boldsymbol{\rho}$ & $\mathbf{V}$ & $\mathbf{E}$ & $\mathbf{F}$ & $\boldsymbol{\rho}^$ & $\mathbf{V}^$ & $\mathbf{E}^$ & $\mathbf{F}^$ \
\hline (i) & 3 & 4 & 6 & 4 & 3 & 4 & 6 & 4 \
\hline (ii) & 3 & 8 & 12 & 6 & 4 & 6 & 12 & 8 \
\hline (iii) & 3 & 20 & 30 & 12 & 5 & 12 & 30 & 20 \
\hline (iv) & 4 & 6 & 12 & 8 & 3 & 8 & 12 & 6 \
\hline (v) & 5 & 12 & 30 & 20 & 3 & 20 & 30 & 12 \
\hline
\end{tabular}

# 数学建模代写

## 数学代写|数学建模代写math modelling代考|Planar Graphs

$$V-E+F=2$$

$$V^-E^+F^*=F-E+V=2$$

## 数学代写|数学建模代写math modelling代考|Regular Solids

$$E=\frac{1}{2} \rho V, F=\frac{\rho}{\rho^*} V$$

$$V-\frac{1}{2} \rho V+\frac{\rho}{\rho *} V=2$$

$$V\left(2 \rho+2 \rho^-\rho \rho^\right)=4 \rho^*$$

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline & $\boldsymbol{\rho}$ & $\mathbf{V}$ & $\mathbf{E}$ & $\mathbf{F}$ & $\boldsymbol{\rho}^$ & $\mathbf{V}^$ & $\mathbf{E}^$ & $\mathbf{F}^$ \hline (i) & 3 & 4 & 6 & 4 & 3 & 4 & 6 & 4 \hline (ii) & 3 & 8 & 12 & 6 & 4 & 6 & 12 & 8 \hline (iii) & 3 & 20 & 30 & 12 & 5 & 12 & 30 & 20 \hline (iv) & 4 & 6 & 12 & 8 & 3 & 8 & 12 & 6 \hline (v) & 5 & 12 & 30 & 20 & 3 & 20 & 30 & 12 \hline
\end{tabular}

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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