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# 数学代写|图论代考GRAPH THEORY代写|The Degree of a Vertex

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## 数学代写|图论代写GRAPH THEORY代考|The Degree of a Vertex

There are many numbers, referred to as parameters, associated with a graph $G$. Knowing the values of certain parameters provides us with information about $G$ but rarely tells us the entire structure of $G$. (These comments are tied in with the concept of isomorphic graphs, which will be discussed in Chapter 3.) We’ve already mentioned the best known parameters: the order and the size. There are also numbers associated with each vertex of a graph. We now consider the best known of these.

The degree of a vertex $v$ in a graph $G$ is the number of edges incident with $v$ and is denoted by $\operatorname{deg}_G v$ or simply by $\operatorname{deg} v$ if the graph $G$ is clear from the context. Also, $\operatorname{deg} v$ is the number of vertices adjacent to $v$. Recall that two adjacent vertices are referred to as neighbors of each other. The set $N(v)$ of neighbors of a vertex $v$ is called the neighborhood of $v$. Thus $\operatorname{deg} v=|N(v)|$.

A vertex of degree 0 is referred to as an isolated vertex and a vertex of degree 1 is an endvertex (or a leaf). The minimum degree of $G$ is the minimum degree among the vertices of $G$ and is denoted by $\delta(G)$; the maximum degree of $G$ is denoted by $\Delta(G)$. So if $G$ is a graph of order $n$ and $v$ is any vertex of $G$, then
$$0 \leq \delta(G) \leq \operatorname{deg} v \leq \Delta(G) \leq n-1$$

## 数学代写|图论代写GRAPH THEORY代考|Regular Graphs

We have already mentioned that $0 \leq \delta(G) \leq \Delta(G) \leq n-1$ for every graph $G$ of order $n$. If $\delta(G)=$ $\Delta(G)$, then the vertices of $G$ have the same degree and $G$ is called regular. If $\operatorname{deg} v=r$ for every vertex $v$ of $G$, where $0 \leq r \leq n-1$, then $G$ is $r$-regular or regular of degree $r$. The only regular graphs of order 4 or 5 are shown in Figure 2.5. There is no 1-regular or 3-regular graph of order 5 , as no graph contains an odd number of odd vertices by Corollary 2.3 .

A 3-regular graph is also referred to as a cubic graph. The graphs $K_4, K_{3,3}$ and $Q_3$ are cubic graphs; however, the best known cubic graph may very well be the Petersen graph, shown in Figure 2.6. We will see this graph again. (Indeed, Section 8.5 is devoted to this graph.)

By Corollary 2.3, there are no $r$-regular graphs of order $n$ if $r$ and $n$ are both odd. However, provided $0 \leq r \leq n-1$, there are no other restrictions on the existence of an $r$-regular graph of order $n$. In the next proof, we will be considering a graph $G$ with vertex set $V(G)=\left{v_1, v_2, \ldots, v_n\right}$ and performing arithmetic on the subscripts of the vertices. We follow the standard practice of performing the arithmetic modulo $n$. For example, if $n=6$ and $i=5$, then the vertex $v_{i+2}$ denotes $v_1$.

## 数学代写|图论代写GRAPH THEORY代考|The Degree of a Vertex

0度的顶点称为孤立顶点，1度的顶点称为终顶点(或叶)。$G$的最小度是$G$的顶点间的最小度，用$\delta(G)$表示;$G$的最大程度用$\Delta(G)$表示。如果$G$是一个阶为$n$的图而$v$是$G$的任意顶点，那么
$$0 \leq \delta(G) \leq \operatorname{deg} v \leq \Delta(G) \leq n-1$$

## 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。