# 数学代写|图论作业代写Graph Theory代考|MATH141

#### Doug I. Jones

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## 数学代写|图论作业代写Graph Theory代考|Decision Trees

Decision trees allow for options to be mapped out, where you can use a series of questions to arrive at the best solution. You have probably seen flow charts used for such a purpose.

Another use for decision trees is to map out the solution space of a game. A current favorite in my house is the game “Guess Who?” where each player draws a card matching one of the 24 people on their board and the goal is to determine which person their opponent has chosen, through a series of Yes/No questions.

To form the tree for this game, we would have a vertex represent the set of people that your opponent could have chosen. This is a rooted tree with the root being the set of all 24 cards. The vertices at level 1 would be the two distinct sets of cards that are available after the first question asked, such as “Does your person have blue eyes?” with one set being those with blue eyes and the other the people without blue eyes. The next level would be the sets formed after the second question, perhaps “Does your person have brown hair?” At this point, we would have 4 distinct sets of people.

Note that the tree formed by this game is a binary tree since at each stage we only have two options for the answer. Also, the order in which the questions are asked would create different trees. When playing the game, you also would not be looking for all possible paths from thè root, only along the branch based on your opponents answer. For example, we could map out the four sets obtained from the two questions above, but in reality would only care about thé unique sêt arrivèd at from how your opponent answered. Thus, the order of your questions could have an impact on how quickly you arrive at a solution.

There have been many editions of “Guess Who?” released by Hasbro games, from the original (which only included 5 women) to newer versions using characters from Disney, Star Wars, and Marvel Comics. The example below is a smaller version using mathematicians as the pool of cards.

## 数学代写|图论作业代写Graph Theory代考|Chemical Graph Theory

Chemical Graph Theory uses concepts from graph theory to obtain results about chemical compounds. In particular, individual atoms in a molecule are represented by vertices and an edge denotes a bond between the atoms. One way to determine the number of isomers for a molecule is to determine the number of distinct graphs that contain the correct type of each atom. For hydrocarbons (molecules only containing carbon and hydrogen atoms) the hydrogen-depleted graph is used since the bonds between the carbon atoms will uniquely determine the locations of the hydrogen atoms.

Below are the only two trees on four vertices. These correspond to the only possible isomers of butane $\left(\mathrm{C}4 \mathrm{H}{10}\right)$, namely n-butane $\left(\mathrm{H}_3 \mathrm{C}\left(\mathrm{CH}_2\right)_2 \mathrm{CH}_3\right)$ and isobutane $\left(\left(\mathrm{H}_3 \mathrm{C}\right)_3 \mathrm{CH}\right)$, whose full molecular forms are displayed below their respective hydrogen-depleted graph. By using graph theory, we can prove no other isomers of butane are possible since no other trees on four vertices exist $[2]$.

While molecules do not need to adhere to a tree structure, for those that do we can use our knowledge of graphs to determine all possible isomers. In doing so, we must keep in mind the chemical properties of the atoms involved. For example, if we restricted ourselves to the hydrogen-depleted graph for hexane $\left(\mathrm{C}6 \mathrm{H}{14}\right)$, we would begin by looking at the possible trees on 6 vertices. Below are two such trees on 6 vertices, but only the one on the left corresponds to an isomer for hexane (called neohaxe or 2,2-dimethylbutane) whereas the tree on the right cannot represent a hexane molecule since the central vertex has degree 5 and carbon atoms have at most 4 bonds. Thus we would need to restrict ourselvés with finding thee different treee structuress with maximum degree 4 (see Exercise 3.28).

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## 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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