# 数学代写|最优化理论作业代写optimization theory代考|CSE276

#### Doug I. Jones

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## 数学代写|最优化理论作业代写optimization theory代考|Basic Definitions

A graph $G$ is a pair $G=(V, E)$ of disjoint finite sets where
$$E \subset\left(\begin{array}{l} V \ 2 \end{array}\right):={e \subset V:|e|=2} .$$
$V=V(G)$ is called the vertex set of $G$, its elements vertices or nodes or points, $E=E(G)$ its edge set. If ${x, y}=e \in E$, we usually omit the brackets and write $e=x y$. In this case, $x$ and $y$ are called the endvertices of $e$. We also say that $x$ and $y$ are adjacent or that they are joined by $e$. The edge $e$ is incident to its endvertices and the number of edges incident to $x \in V$ is called the degree of $x, d(x)=d_G(x)$. A graph $G$ with $E(G)=\left(\begin{array}{c}V(G) \ 2\end{array}\right)$ is called complete. If $|V(G)|=n$, it is denoted by $K_n$. $G$ is called $r$-regular if all degrees equal $r$.

Graphs are visualized by drawing diagrams such that the vertices correspond to distinguished points in the plane and two such points are joined by a line if and only if the corresponding vertices are adjacent. We emphasize that Graph-theoretical terminology is still far from being unified. Here we essentially follow Bollobás, whose introductory text is highly recommended (see [27]).

Some authors allow different edges joining the same endpoints (multiple edges) and/or edges joining a vertex to itself (loops). We call such objects multigraphs, but they will not appear too often in this book. Usually, the definitions we give for graphs carry over to multigraphs in an obvious way.
Two graphs $G$ and $G^{\prime}$ are isomorphic $\left(G \simeq G^{\prime}\right)$ iff there exists a bijection $\Phi: V(G) \rightarrow V\left(G^{\prime}\right)$ such that
$$x y \in E(G) \Longleftrightarrow \Phi(x) \Phi(y) \in E\left(G^{\prime}\right) .$$

## 数学代写|最优化理论作业代写optimization theory代考|Matchings

The first graph-theoretical topic we are going to study in depth is Matching Theory. Let us motivate the concept by two examples:

Example 14.2.1 In a factory, there are $n$ workers and $m$ jobs. Each worker can do only one job at a time and each job needs only one worker. According to different qualifications, not every worker can do every job but we know for each worker the set of jobs for which she or he is qualified. We wish to maximize the number of jobs which can be done simultaneously by some qualified workers.

To formalize this problem, we first introduce a bipartite graph with one colour class consisting of the workers $(W)$ and the other of the jobs $(U)$. Worker $w_i$ and job $u_j$ are adjacent if and only if $w_i$ is qualified for $u_j$. An assignment of workers to jobs for which they are qualified is now represented by a set $M$ of edges in the bipartite graph such that each worker $w_i$ and each job $u_j$ is an endpoint of at most one edge in $M$. Such a set of edges is called a matching.

Definition 14.2.2 Suppose $G=(V, E)$ is a graph and $M \subset E$. $M$ is called a matching if each vertex in $V$ is incident to at most one edge in $M$. In this case, the edges in $M$ are called independent. We denote by $\nu(G)$ the maximum cardinality of a matching in $G$. Any matching $M$ with $|M|=\nu(G)$ is called a maximum matching. The vertices which are incident to matching edges are said to be covered (by $M$ ). The other vertices are called exposed (with respect to $M$ ). A matching covering all the nodes of $G$ is called a perfect matching.

Note that there are matchings which are maximal with respect to inclusion but have fewer edges than $\nu(G)$. Such matchings are called maximal (not maximum). In our example, the graph we constructed was bipartite, corresponding to the natural partition of the “vertices” into workers and jobs. This is in fact an important special case but the general matching problem also arises in applications.

# 最优化代写

## 数学代写|最优化理论作业代写optimization theory代考|Basic Definitions

$$E \subset\left(\begin{array}{l} V \ 2 \end{array}\right):={e \subset V:|e|=2} .$$
$V=V(G)$被称为$G$的顶点集，它的元素顶点或节点或点，$E=E(G)$它的边集。如果是${x, y}=e \in E$，我们通常省略括号，写成$e=x y$。在这种情况下，$x$和$y$被称为$e$的端点。我们也说$x$和$y$相邻或者它们被$e$连接。边$e$与它的顶点相关联，与$x \in V$相关联的边的数量称为$x, d(x)=d_G(x)$的度。带有$E(G)=\left(\begin{array}{c}V(G) \ 2\end{array}\right)$的图$G$称为完整图。如果是$|V(G)|=n$，则用$K_n$表示。如果所有度都等于$r$，则$G$称为$r$ -regular。

$$x y \in E(G) \Longleftrightarrow \Phi(x) \Phi(y) \in E\left(G^{\prime}\right) .$$

## 数学代写|最优化理论作业代写optimization theory代考|Matchings

14.2.2假设$G=(V, E)$是一个图，$M \subset E$。如果$V$中的每个顶点最多与$M$中的一条边关联，则称$M$为匹配。在这种情况下，$M$中的边被称为独立边。我们用$\nu(G)$表示$G$中匹配的最大基数。$M$与$|M|=\nu(G)$的任何匹配都称为最大匹配。与匹配边相关的顶点被称为覆盖(通过$M$)。其他顶点称为暴露(相对于$M$)。覆盖$G$所有节点的匹配称为完美匹配。

## 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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