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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|图论作业代写Graph Theory代考|Walks Using Matrices

Recall in Section $1.4$ we saw how to model a graph using an adjacency matrix. Matrix representations of graphs are useful when using a computer program to investigate certain features or processes on a graph. Another use for the adjacency matrix is to count the number of walks between two vertices within a graph. For review of matrix operations, see Appendix C.

Consider the graph shown below with its adjacency matrix $A$ on the right.

If we want a walk of length 1 , we are in essence asking for an edge between two vertices. So to count the number of walks of length 1 from $v_1$ to $v_3$, we need only to count the number of edges (namely 2) between these vertices. What if we want the walks of length 2 ? By inspection, we can see there is only one, which is
$$v_1 \underset{e_3}{\rightarrow} v_2 \underset{e_4}{\rightarrow} v_3$$
Now consider the walks from $v_1$ to $v_2$. There is only one walk of length 1 , and yet three of length 2 :
\begin{aligned} &v_1 \underset{e_3}{\rightarrow} v_2 \underset{e_5}{\rightarrow} v_2 \ &v_1 \underset{e_1}{\rightarrow} v_3 \underset{e_4}{\rightarrow} v_2 \ &v_1 \underset{e_2}{\rightarrow} v_3 \underset{e_4}{\rightarrow} v_2 \end{aligned}
How could we count this? If we know how many walks there are from $v_1$ to $v_2$ (1) and then the number from $v_2$ to itself $(1)$, we can get one type of walk from $v_1$ to $v_2$. Also, we could count the number of walks from $v_1$ to $v_3(2)$ and then the number of walks from $v_3$ to $v_2$ (1). In total we have $1 * 1+2 * 1=3$ walks from $v_1$ to $v_2$. Note that we did not include any walks of the form $v_1 v_1 v_2$ since there are no edges from $v_1$ to itself.
Viewing this as a multiplication of vectors, we have
$$\left[\begin{array}{lll} 0 & 1 & 2 \end{array}\right] \cdot\left[\begin{array}{l} 1 \ 1 \ 1 \end{array}\right]=0 * 1+1 * 1+2 * 1=3$$
If we do this for the entire adjacency matrix, we have
$$A^2=\left[\begin{array}{lll} 5 & 3 & 1 \ 3 & 3 & 3 \ 1 & 3 & 5 \end{array}\right]$$

## 数学代写|图论作业代写Graph Theory代考|Distance, Diameter, and Radius

Dijkstra’s Algorithm provides the method for determining the shortest path between two points on a graph, which we define as the distance between those vertices. There are many theoretical implications for this distance. We will investigate a few of these below; further discussion will occur throughout this book, most notably in Chapter 3 when discussing trees and in Chapter 4 when discussing connectivity. In particular, we will begin with defining the diameter and radius of a graph and the eccentricity of a vertex.

Definition 2.25 Given two vertices $x, y$ in a graph $G$, define the distance $d(x, y)$ as the length of the shortest path from $x$ to $y$. The eccentricity of a vertex $x$ is the maximum distance from $x$ to any other vertex in $G$; that is $\epsilon(x)=\max _{y \in V(G)} d(x, y)$.

The diameter of $G$ is the maximum eccentricity among all vertices, and so measures the maximum distance between any two vertices; that is $\operatorname{diam}(G)=\max {x, y \in V(G)} d(x, y)$. The radius of a graph is the minimum eccentricity among all vertices; that is $\operatorname{rad}(G)=\min {x \in V(G)} \epsilon(x)$.

If a graph is connected, all of these parameters will be nonnegative integers. What happens if the graph is disconnected? If $x$ and $y$ are in separate components of $G$ then there is no shortest path between them and $d(x, y)=\infty$. This would then make $\operatorname{diam}(G)=\operatorname{rad}(G)=\infty$ since $\epsilon(v)=\infty$ for all vertices in $G$. Conceptually, you can think of the diameter as the longest path you can travel between any two points on a graph and the radius as the shortest distance among all pairs of vertices.

# 图论代考

## 数学代写|图论作业代写Graph Theory代考|Walks Using Matrices

$$v_1 \underset{e_3}{\rightarrow} v_2 \underset{e_4}{\rightarrow} v_3$$

$$v_1 \underset{e_3}{\rightarrow} v_2 \underset{e_5}{\rightarrow} v_2 \quad v_1 \underset{e_1}{\rightarrow} v_3 \underset{e_4}{\rightarrow} v_2 v_1 \underset{e_2}{\rightarrow} v_3 \underset{e_4}{\rightarrow} v_2$$

$$\left[\begin{array}{lll} 0 & 1 & 2 \end{array}\right] \cdot\left[\begin{array}{lll} 1 & 1 & 1 \end{array}\right]=0 * 1+1 * 1+2 * 1=3$$

$$A^2=\left[\begin{array}{llllllll} 5 & 3 & 13 & 3 & 31 & 3 & 5 \end{array}\right]$$

## 数学代写|图论作业代写Graph Theory代考|Distance, Diameter, and Radius

Dijkstra 算法提供了确定图上两点之间最短路径的方法，我们将其定义为这些顶点之间的距离。这个距离有很多理论含义。我们将在下面调查其中的一些；本书将进一步讨论，尤其是在第 3 章讨论树时和第 4 章讨论连通性时。特别是，我们将从定义图形的直径和半径以及顶点的偏心率开始。

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师