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# 数学代写|组合学代写Combinatorics代考|Spanning Trees

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## 数学代写|组合学代写Combinatorics代考|Spanning Trees

Suppose we wish to install “lines” to link various sites together. A site may be a computer installation, a town or a spy. A line may be a digital communication channel, a rail line or a contact arrangement. We’ll assume that

• a line operates in both directions;
• it must be possible to get from any site to any other site using lines;
• each possible line has a cost (rental rate, construction costs or likelihood of detection) independent of each other line’s cost;
• we want to choose lines to minimize the total cost.
We can think of the sites as vertices $V$ in a graph, the lines as edges $E$ and the costs as a function $\lambda$ from the edges to the real numbers. Let $T=\left(V, E^{\prime}\right)$ be a subgraph of $G=(V, E)$. Define $\lambda(T)$, the weight of $T$, to be the sum of $\lambda(e)$ over all $e \in E^{\prime}$. Minimizing total cost means choosing $T$ so that $\lambda(T)$ is a minimum. Getting from one site to another means choosing $T$ so that it is connected. It follows that we should choose $T$ to be a spanning tree-if $T$ had more edges than in a spanning tree, we could delete some; if $T$ had less, it would not be connected. (See Exercise 5.4.2 (p. 139).) We call such a $T$ a minimum weight spanning tree of $(G, \lambda)$, or simply of $G$, with $\lambda$ understood from context.

## 数学代写|组合学代写Combinatorics代考|Lineal Spanning Trees

If we simply want to find any spanning tree of $G$, we can choose any values for the function $\lambda$, and use the minimal weight spanning tree algorithm of Theorem 6.1. Put another way, in Step 3 we may choose any edge $f \in F$. Sometimes it is important to restrict the choice of $f$ in some way so that the spanning tree will have some special property other than being minimal.

An important example of such a special property concerns certain rooted spanning trees. To define the trees we are interested in, we borrow some terminology from genealogy.

Definition 6.2 Lineal spanning tree Let $x$ and $y$ be two vertices in a rooted tree with root $r$. If $x$ is on the path connecting $r$ to $y$, we say that $y$ is a descendant of $x$. (In particular, all vertices are descendants of $r$.) If one of $u$ and $v$ is a descendant of the other, we say that ${u, v}$ is a lineal pair. A lineal spanning tree or depth first spanning tree of a connected graph $G=(V, E)$ is a rooted spanning tree of $G$ such that each edge ${u, v}$ of $G$ is a lineal pair.
To see some examples of a lineal spanning tree, look back at Figure 5.5 (p. 139). It is the lineal spanning tree of a graph, namely itself. We can add some edges to this graph, for example ${a, f}$ and ${b, j}$ and still have Figure 5.5 as a lineal spanning tree. On the other hand, if we added the edge ${e, j}$, the graph would not have Figure 5.5 as a lineal spanning tree.

How can we find a lineal spanning tree of a graph? That question may be a bit premature-we don’t even know when such a tree exists. We’ll prove

Theorem 6.2 Lineal spanning tree existence Every connected graph $G$ has a lineal spanning tree. In fact, given any vertex $r$ of $G$, there is a lineal spanning tree of $G$ with root $r$.

## 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。