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# 数学代写|图论代写Graph Theory代考|Graph representations

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## 数学代写|图论代写Graph Theory代考|Data structures

There are different ways to represent graphs. Perhaps the most appealing one is to use an adjacency matrix. Consider a graph $G$ with $n$ vertices and $m$ edges. Its adjacency matrix is nothing else but a table $\mathbf{A}$ with $n$ rows and $n$ columns with entry $\mathbf{A}[i, j]$ denoting the number of edges joining vertex $v_{i}$ and $v_{j}$. To illustrate, Figure $2.7$ shows a simple graph with its accompanying adjacency matrix.
It is not difficult to see that the following properties hold:

• An adjacency matrix is symmetric, that is for all $i, j, \mathbf{A}[i, j]=\mathbf{A}[j, i]$. This property reflects the fact that an edge is represented as an unordered pair of vertices $e=\left\langle v_{i}, v_{j}\right\rangle=\left\langle v_{j}, v_{i}\right\rangle$.
• A graph $G$ is simple if and only if for all $i, j, \mathbf{A}[i, j] \leq 1$ and $\mathbf{A}[i, i]=0$. In other words, there can be at most one edge joining vertices $v_{i}$ and $v_{j}$ and, in particular, no edge joining a vertex to itself.
• The sum of values in row $i$ is equal to the degree of vertex $v_{i}$, that is, $\delta\left(v_{i}\right)=\sum_{j=1}^{n} \mathbf{A}[i, j] .$

## 数学代写|图论代写Graph Theory代考|Graph isomorphism

An important observation is that all these representations are independent of the way that we draw a graph. Consider the graphs shown in Figure 2.9. No matter whether we represent each graph by its adjacency matrix, incidence matrix, or edge list, if we properly attach labels to vertices and edges, we will find that their respective representations are exactly the same. As a consequence, they should also be considered to be the same. This notion of similarity is formalized through what is known as graph isomorphism.
Definition 2.7: Consider two graphs $G=(V, E)$ and $G^{}=\left(V^{}, E^{}\right)$. $G$ and $G^{}$ are isomorphic if there exists a one-to-one mapping $\phi: V \rightarrow V^{}$ such that for every edge $e \in E$ with $e=\langle u, v\rangle$, there is a unique edge $e^{} \in E^{}$ with $e^{}=$ $\langle\phi(u), \phi(v)\rangle$.

Stated differently, two graphs $G$ and $G^{}$ are isomorphic if we can uniquely map the vertices and edges of $G$ to those of $G^{}$ such that if two vertices were joined in $G$ by a number of edges, their counterparts in $G^{*}$ will be joined by the same number of edges.

## 数学代写|图论代写GRAPH THEORY代考|DATA STRUCTURES

• 邻接矩阵是对称的，即对所有一世,j,一种[一世,j]=一种[j,一世]. 该属性反映了一条边被表示为一对无序顶点的事实和=⟨在一世,在j⟩=⟨在j,在一世⟩.
• 图表G简单当且仅当对所有人一世,j,一种[一世,j]≤1和一种[一世,一世]=0. 换句话说，最多可以有一条边连接顶点在一世和在j尤其是没有将顶点连接到自身的边。
• 行中的值的总和一世等于顶点的度数在一世， 那是，$i$ is equal to the degree of vertex $v_{i}$, that is, $\delta\left(v_{i}\right)=\sum_{j=1}^{n} \mathbf{A}[i, j] .$

## 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。