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## 数学代写|图论代写GRAPH THEORY代考|Mathematical Induction

The last proof technique we review is quite useful when studying discrete objects, especially objects that can easily be transformed into ones of smaller size. Mathematical induction relies on a two step process. In the first step (sometimes referred to as the base case or basis step) we show the statement to be proved holds for a specific value or size. In the second step (called the induction step) we assume that the statement holds for some unknown value and then show the statement also holds for the next value.

The power of induction is that we are proving a statement that holds for an infinite number of objects but only need to prove two very specific items. As we will see throughout this book, graphs naturally lend themselves to induction proofs by our ability to take a graph of a specific size (usually in terms of the number of vertices or edges) and make it smaller by removing either an edge or a vertex. This technique is shown in the result below about the number of edges in a complete graph.
Proposition 1.25 The complete graph $K_n$ has $\frac{n(n-1)}{2}$ edges.

## 数学代写|图论代写GRAPH THEORY代考|Degree Sequence

As noted above, the degree of a vertex can be seen in the adjacency matrix for a graph by summing the entries along its corresponding row or column. Here we switch gears a bit to look at what conditions must be placed on a sequence of integers so that they could be the degrees of a simple graph.
Definition 1.26 The degree sequence of a graph is a listing of the degrees of the vertices. It is customary to write these in decreasing order. If a sequence is a degree sequence of a simple graph then we call it graphical.

Example 1.18 Explain why neither 4,4,2,1,0 nor 4,4,3,1,0 can be graphical.

Solution: The first sequence sums to 11 , but we know the sum of the degrees of a graph must be even by the Handshaking Lemma. Thus it cannot be a degree sequence.

The second sequence sums to 12 , so it is at least even. However, in a simple graph with 5 vertices, a vertex with degree 4 must be adjacent to all the other vertices, which would mean no vertex could have degree 0 . Thus the second sequence cannot be a degree sequence.

## 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。

Posted on Categories:Graph Theory, 图论, 数学代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|图论代写GRAPH THEORY代考|Partial Intuitionistic Fuzzy Labeling Tree

Partial IFLT is another type of labeling tree.
Definition $9.45$ A connected IFLG $\mathscr{G}$ is said to be a partial IFLT if $\mathscr{G}$ has a spanning subgraph $F$ that is a tree and for every $\operatorname{arc}(p, q)(\notin F)$ of $\mathscr{G}, C O N N_{1 \mathscr{G}}(p, q)>$ $\mu_1(p, q)$ and $C O N N_{2 S}(p, q)<\mu_2(p, q)$.

When a graph $\mathscr{G}$ is disconnected and the above condition holds for all components of $\mathscr{G}$, then $\mathscr{G}$ is called a partial fuzzy forest.
Following is a characterization of partial IFLT.
Theorem $9.33$ A connected IFLG $\mathscr{G}$ is a partial IFLT if and only if for any cycle $\mathscr{C}$ in $\mathscr{G}$, there is an arc $e=(p, q)$ with $\mu_1(e)$ CONN $\mathrm{N}_{2(\mathscr{S}-e)}(p, q)$.

Proof Suppose $\mathscr{G}$ is a connected IFLG. If there is no cycle, then $\mathscr{G}$ is obviously a tree and also a partial fuzzy tree. If there is cycle in $\mathscr{G}$, let $(p, q)$ be an arc of $\mathscr{C}$ with the least membership and greatest non-membership values in $\mathscr{G}$. Remove the arc $(p, q)$ from $\mathscr{G}$. If $\mathscr{G}$ has another cycle, repeat the process. Not at each step no previously removed arc is strongest the arc being presently deleted. When the graph $\mathscr{G}$ does not contain any cycle, then the subgraph is a tree $F$. Suppose the $\operatorname{arc}(p, q)$ is not in $F$. Then $(p, q)$ is one of the arcs removed in the process to construct $F$. Since $F$ is a tree and $(p, q)$ is the arc having least membership and greatest non-membership values among all the arcs of a cycle in $\mathscr{G}$, it follows that there is a path between $a$ and $b$ whose membership value is greater than $\mu_1(p, q)$ and non-membership value is less than $\mu_2(p, q)$ and this does not cover $(p, q)$ or any other edges removed before. If that path covers arcs that were removed later, then the path can be further detached and so on. This process gives a path belonging entirely the arcs of $F$. Thus $\mathscr{G}$ is a partial IFLT.

## 数学代写|图论代写GRAPH THEORY代考|Partial Intuitionistic Fuzzy Labeling Tree

Partial IFLT is another type of labeling tree.
Definition $9.45$ A connected IFLG $\mathscr{G}$ is said to be a partial IFLT if $\mathscr{G}$ has a spanning subgraph $F$ that is a tree and for every $\operatorname{arc}(p, q)(\notin F)$ of $\mathscr{G}, C O N N_{1 \mathscr{G}}(p, q)>$ $\mu_1(p, q)$ and $C O N N_{2 \mathscr{G}}(p, q)<\mu_2(p, q)$.

When a graph $\mathscr{G}$ is disconnected and the above condition holds for all components of $\mathscr{G}$, then $\mathscr{G}$ is called a partial fuzzy forest.
Following is a characterization of partial IFLT.
Theorem $9.33$ A connected IFLG $\mathscr{G}$ is a partial IFLT if and only if for any cycle $\mathscr{C}$ in $\mathscr{G}$, there is an arc $e=(p, q)$ with $\mu_1(e)$ $C O N N_{2(\mathscr{S}-e)}(p, q)$.

Proof Suppose $\mathscr{G}$ is a connected IFLG. If there is no cycle, then $\mathscr{G}$ is obviously a tree and also a partial fuzzy tree. If there is cycle in $\mathscr{G}$, let $(p, q)$ be an arc of $\mathscr{C}$ with the least membership and greatest non-membership values in $\mathscr{G}$. Remove the arc $(p, q)$ from $\mathscr{G}$. If $\mathscr{G}$ has another cycle, repeat the process. Not at each step no previously removed arc is strongest the arc being presently deleted. When the graph $\mathscr{G}$ does not contain any cycle, then the subgraph is a tree $F$. Suppose the arc $(p, q)$ is not in $F$. Then $(p, q)$ is one of the arcs removed in the process to construct $F$. Since $F$ is a tree and $(p, q)$ is the arc having least membership and greatest non-membership values among all the arcs of a cycle in $\mathscr{G}$, it follows that there is a path between $a$ and $b$ whose membership value is greater than $\mu_1(p, q)$ and non-membership value is less than $\mu_2(p, q)$ and this does not cover $(p, q)$ or any other edges removed before. If that path covers arcs that were removed later, then the path can be further detached and so on. This process gives a path belonging entirely the arcs of $F$. Thus $\mathscr{G}$ is a partial IFLT.

Conversely, suppose $\mathscr{G}$ is a partial IFLT and $\mathscr{P}$ is cycle, then for some arc $e=$ $(p, q)$ of $\mathscr{P}$ does not lie on $F$. Thus, by definition, $\mu_1(e)C O N N_{2(\mathscr{G}-e)}(p, q)>C O N N_{2 \mathscr{}}(p, q)$

## 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。