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## 数学代写|图论代写GRAPH THEORY代考|Connectivity and Paths

Now that we have some familiarity with connectivity, we turn to its relationship to paths within a graph. Note that for the remainder of this section, we will assume the graphs are connected, as otherwise the results are trivial. We begin by relating cut-vertices and bridges to paths. You should notice that almost every result for vertices has an edge analog. We begin with the most simple results relating a cut-vertex or a bridge to its presence on a path.
Theorem 4.6 A vertex $v$ is a cut-vertex of a graph $G$ if and only if there exist vertices $x$ and $y$ such that $v$ is on every $x-y$ path.
Proof: First suppose $v$ is a cut-vertex in a graph $G$. Then $G-v$ must have at least two components. Let $x$ and $y$ be vertices in different components of $G-v$. Since $G$ is connected, we know there must exist an $x-y$ path in $G$ that does not exist in $G-v$. Thus $v$ must lie on this path.

Conversely, let $v$ be a vertex and suppose there exist vertices $x$ and $y$ such that $v$ is on every $x-y$ path. Then none of these paths exist in $G-v$, and so $x$ and $y$ cannot be in the same component of $G-v$. Thus $G$ must have at least two components and so $v$ is a cut-vertex.

## 数学代写|图论代写GRAPH THEORY代考|Menger’s Theorem

The following theorems generalize the results above relating a cut-vertex or bridge to paths in a graph. Menger’s Theorem, and the resulting theorems, show the number of internally disjoint (or edge-disjoint) paths directly corresponds to the connectivity (or edge-connectivity) of a graph. For example, in $G_2$ above we could separate $b$ and $c$ using two vertices and it should be easy to see that $b h c$ and $b e f d c$ are internally disjoint $b-c$ paths. However, if we try to find more than two $b-c$ paths then one of them cannot be internally disjoint from the others (try it!).

Theorem 4.13 (and its edge analog) is named for Karl Menger, the Austrian-American mathematician who first published the result in 1927 [65]. There are many different versions of the proof, and the one presented here

most closely resembles that in [21]. Note that for this proof we need an additional process, call a contraction.

Definition 4.12 Let $e=x y$ be an edge of a graph $G$. The contraction of $e$, denoted $G / e$, replaces the edge $e$ with a vertex $v_e$ so that any vertices adjacent to either $x$ or $y$ are now adjacent to $v_e$. Contracting an edge creates a smaller graph, both in terms of the number of vertices and edges, but keeps much of the structure of a graph in tact. In particular, contracting an edge cannot disconnect a graph (see Exercise 4.22).

## 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

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## 数学代写|图论代写GRAPH THEORY代考|Rooted Trees

When we look at the graph theory terminology used for trees, mathematicians have adopted many of the same terms that we use for biological trees (such as leaf and forest). So what then would we mean by a root of a tree? We can think of the root as the place from which a tree grows.

Definition 3.15 A rooted tree is a tree $T$ with a special designated vertex $r$, called the root. The level of any vertex in $T$ is defined as the length of its shortest path to $r$. The height of a rooted tree is the largest level for any vertex in $T$.

Example 3.6 Find the level of each vertex and the height of the rooted tree shown below.

Solution: Vertices $a$ and $b$ are of level $1, c, d, e$, and $f$ of level 2 , and $g$ and $h$ of level 3 . The root $r$ has level 0 . The height of the tree is 3 .

Most people have encountered a specific type of rooted tree: a family tree. In fact, much of the terminology for rooted trees comes not from a plant version of a tree but rather from genealogy and family trees. The root of a family tree would be the person for whom the descendants are being mapped and the level of a vertex would represent a generation; see the tree below. With this application in mind, the terminology below is used to describe how various vertices are related within a rooted tree.

## 数学代写|图论代写GRAPH THEORY代考|Depth-First Search Tree

The main idea behind a depth-first tree is to travel along a path as far as possible from the root of a given graph. If this path does not encompass the entire graph, then branches are built off this central path to create a tree. The formal description of this algorithm relies on an ordered listing of the neighbors of each vertex and uses this order when adding new vertices to the tree. For simplicity, we will always use an alphabetical order when considering neighbor lists.
Depth-First Search Tree
Input: Simple graph $G=(V, E)$ and a designated root vertex $r$.
Steps:

Choose the first neighbor $x$ of $r$ in $G$ and add it to $T=\left(V, E^{\prime}\right)$.

Choose the first neighbor of $x$ and add it to $T$. Continue in this fashion-picking the first neighbor of the previous vertex to create a path $P$. If $P$ contains all the vertices of $G$, then $P$ is the depth-first search tree. Otherwise continue to Step (3).

Backtrack along $P$ until the first vertex is found that has neighbors not in $T$. Use this as the root and return to Step (1).
Output: Depth-first search tree $T$.

## 数学代写图论代写GRAPH THEORY代考|Rooted Trees

Example 3.6 找到每个顶点的层数和下图所示的有根树的高度。

## 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

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## 数学代写|图论代写GRAPH THEORY代考|Chromatic Index of Fuzzy Graph

The least number of basic colors used to color a FG is called the chromatic index (CI) of a FG. Suppose, such least number of basic colors be $N$. This CI is not sufficient to mention the strengths of the edges. So, we redefined the CI as a number with two components, say ( $N, W)$, where $W$ is the weight and we call it fuzzy CI. The weight is defined as
$$W=\sum_{i=1}^N\left{\max j f{e_j}\left(c_i\right)\right},$$
where the basic color $c_i$ is assigned to the edge $e_j$ for some $j$, and the intensity (or membership value) of the color $c_i$ is $f_{e_j}\left(c_i\right)$. This weight is meaningful only when it is very high or very low value, i.e. every edge is strong, or every edge is weak. Therefore, the weights need further restrictions.

Example 7.3 Let us consider the graph of Fig. 7.6. In this FG, three basic colors red, black, and green are assigned to the vertices, i.e. for this case $N=3$. The intensity of colors is the strength of the corresponding edge. The red color is assigned to the edges $C A$ and $B D$ with membership values $0.5$ and $0.83$ respectively. The black color is assigned to the edges $B C$ and $A D$ with membership value 1 . The green color is given to the edges $C D$ and $A B$ with membership values 1 and $0.71$ respectively. Therefore, the weight $W=0.83+1+1=2.83$. Hence, CI of this FG is $(3,2.83)$.
Lemma 7.2 The CI of a complete fuzzy graph is $(N, N)$.
The upper value of the weight is stated below.
Lemma 7.3 Let $(N, W)$ be the CI of a FG, then $0<W \leq N$.

## 数学代写|图论代写GRAPH THEORY代考|Strong Chromatic Index of Edge Coloring of Fuzzy Graph

The weight of the fuzzy CI is significant only if the weight is very large or very small. Suppose $(3,2.9)$ and $(3,0.3)$ are fuzzy CIs of two FGs. From these CIs, one can conclude that the edges of the second graph are not strong. But, if the weight is near about half of its upper bound (i.e. $N / 2$ ), then it does not make any clear conclusion. So, we need a modification of $\mathrm{CI}$ and hence strong $\mathrm{CI}$ is defined to explain such case. Let $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$ be a connected FG. To color the edges of a FG sometimes we use fuzzy colors whose membership values are more than $0.5$. Such colors are called strong colors and this leads us to define strong CI. The strong CI is denoted by $\gamma_s(\mathscr{G})=\left(M_s, W_s\right)$, where $M_s$ is the number of basic colors required to color $\mathscr{G}$ and $W_s$ is the sum of membership values of the basic colors.

Example 7.4 Let us consider a FG whose set of vertices and edges be ${a(0.7), b(0.5)$, $c(0.4), d(0.6), e(0.8)}$ and ${a b(0.4), a c(0.1), a d(0.2), a e(0.7)}$. In this graph, four basic colors, viz. red, yellow, green, and blue are used to color the edges, and the membership values of the colors are $0.8,0.25,0.33,1$ respectively (calculated by $\left.\frac{\mu(x, y)}{\sigma(x) \wedge \sigma(y)}\right)$. Thus, $M_s=2$ and $W_s=(0.8+1)=1.8$. Hence, the strong CI of this FG is $(2,1.8)$.

Theorem 7.7 Let $\mathscr{G}$ be a FG, and the CI and strong CI of $\mathscr{G}$ be $(N, W)$ and $\left(M_s, W_s\right)$, then
(i) $N \geq M_s$ and $W \geq W_s$,
(ii) $2 W_s-M$ is either zero or positive.
Theorem 7.8 Let $\mathscr{G}$ be a FG and its strong CI be $\left(M_s, W_s\right)$. Then $\frac{M_s}{2} \leq W_s \leq M_s$ is true.

Proof Let $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$ be a FG and its strong $\mathrm{CI}$ be $\left(M_s, W_s\right)$. Therefore, the $\mathrm{FG} \mathscr{G}$ is colored by $M_s$ number of strong basic color and the membership value of each such strong basic colors is at least $0.5$. Thus, $W_s=\left{0.5+0.5+\cdots M_s\right.$ times $}=\frac{M_s}{2}$.
So, the least value of strong weight is $\frac{M_s}{2}$. Also, the maximum intensity of a color is 1 . So, $W_s \leq{1+1+1+\cdots M$ times $}=M_s$. Hence, the result.

## 数学代写|图论代写GRAPH THEORY代考|模糊图的色度指数

$$W=sum_{i=1}^N\left{max j f{e_j}\left(c_i\right)\right}。$$

## 数学代写|图论代写GRAPH THEORY代考|模糊图的边缘着色的强色度指数

(i) $N\geq M_s$和$W\geq W_s$。
(ii) $2 W_s-M$是零或正。

## 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。