## 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 the 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 the unique set arrived 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 ourselves with finding the different tree structures with maximum degree 4 (see Exercise 3.28).

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

“猜猜谁？”有很多版本。由孩之宝游戏发行，从原版（仅包括 5 名女性）到使用迪斯尼、星球大战和漫威漫画人物的新版本。下面的示例是使用数学家作为卡片池的较小版本。

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

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