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# 数学代写|图论代考GRAPH THEORY代写|Math781 Chromatic Index of Fuzzy Graph

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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。