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数学代写|图论代考GRAPH THEORY代写|MATH2069 m-Step Fuzzy Neighborhood Graph

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数学代写|图论代写GRAPH THEORY代考|m-Step Fuzzy Neighborhood Graph

Here, we describe a particular case of $m$-step fuzzy graph called $m$-step fuzzy nbd graph.

Definition 4.15 Fuzzy $m$-step nbd of a vertex $b$ in a $\mathrm{FG} \mathscr{G}=(\mathscr{V}, \sigma, \mu)$ is a fuzzy set $\mathscr{N}m(b)=\left(X_b, \rho_b\right)$, where $X_b={a \mid$ there exists a fuzzy path from $b$ to $a$ of length $\left.m, P{b, a}^m\right}$ and $\rho_b: X_b \rightarrow[0,1]$ is defined by $\rho_b(a)=\min {\mu(c, d),(c, d)$ is an edge of $P_{b, a}^m$.

Definition 4.16 For the FG $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$, the $m$-step fuzzy nbd graph is denoted by $N_m(\mathscr{G})$ and defined by $N_m(\mathscr{G})=(\mathscr{V}, \sigma, \eta)$, where $\eta: \mathscr{V} \times \mathscr{V} \rightarrow[0,1]$ is the membership value of the edge $(a, b)$ and is given by $\eta(a, b)=\sigma(a) \wedge$ $\sigma(b) h\left(\mathscr{N}_m(a) \cap \mathscr{N}_m(b)\right)$ for all $a, b \in \mathscr{V}$

Theorem 4.11 Let $\overrightarrow{\mathscr{G}}$ be a fuzzy digraph without any parallel edge. Then $C_m(\overrightarrow{\mathscr{G}})=$ $N_m(\mathscr{G})$ for $m>1$, where $\mathscr{G}$ is the underlying $F G$ of $\overrightarrow{\mathscr{G}}$.

Proof Let $\overrightarrow{\mathscr{G}}=(\mathscr{V}, \sigma, \vec{\mu})$ be a fuzzy digraph without parallel edges. Let $\mathscr{G}=$ $(\mathscr{V}, \sigma, \mu)$ be the underlying FG of $\overrightarrow{\mathscr{G}}$. Also, let $C_m(\overrightarrow{\mathscr{G}})=(\mathscr{V}, \sigma, v)$ be the $m$-step FCompG and $N_m(\mathscr{G})=(\mathscr{V}, \sigma, \eta)$ be the $m$-step fuzzy nbd graph. Since $\overrightarrow{\mathscr{G}}$ has no parallel edges, $\vec{\mu}(a, b)=\mu(a, b)$ for all vertices $a, b \in \mathscr{V}$.
Now, we will show that $v(a, b)=\eta(a, b)$ for all edges $(a, b)$.
Since $N_m^{+}(a) \cap N_m^{+}(b)$ in $\overrightarrow{\mathscr{G}}$ is equal to $N_m(a) \cap N_m(b)$ in $\mathscr{G}$ and $v(a, b)=$ $\sigma(a) \wedge \sigma(b) \quad h\left(N_m^{+}(a) \cap N_m^{+}(b)\right), \quad \eta(a, b)=\sigma(a) \wedge \sigma(b) \quad h\left(N_m(a) \cap N_m(b)\right)$. Hence, $v(a, b)=\eta(a, b)$.

数学代写|图论代写GRAPH THEORY代考|Fuzzy Economic Competition Graph

Nowadays, people are transferring money online from different sources (e.g. banks, ATMs, POSs, PayPal, Paytm, etc.) to different destinations within a very short time. To explain this graph, let us consider three funding agencies $F_1, F_2, F_3$ for running scientific projects. Also, it is assumed that five projects say $P_1, P_2, \cdots, P_5$ are running in five different institutions, but one same project may be funded by more than one funding agencies (see Fig. 4.7). The funding agencies and projects are considered as vertices of the graph and there is a directed edge from the funding agency to the project (e.g. the funding agencies for the project $P_1$ are $F_1$ and $F_2$, so there are directed edges $\left(F_1, P_1\right)$ and $\left(F_2, P_1\right)$, and so on $)$.

This section is based on the work by Samanta et al. [14] in 2015. Here, the fuzzy economic competition graph and $m$-step fuzzy economic competition graph are introduced.

Definition 4.17 Let $\overrightarrow{\mathscr{G}}=(\mathscr{V}, \sigma, \vec{\mu})$ be a fuzzy digraph and $E(\overrightarrow{\mathscr{G}})$ be its corresponding fuzzy economic competition graph. $E(\overrightarrow{\mathscr{G}})$ is an undirected FG $\mathscr{G}=(\mathscr{V}, \sigma, \phi)$ which has an edge between two vertices $a, b \in \mathscr{V}$ in $E(\overrightarrow{\mathscr{G}})$ if and only if $\mathscr{N}^{-}(a) \cap \mathscr{N}^{-}(b) \neq \emptyset$ in $\overrightarrow{\mathscr{G}}$ and the membership value $\phi(a, b)$ of the edge $(a, b)$ in $\mathscr{C}(\overrightarrow{\mathscr{G}})$ is given by $\phi(a, b)=(\sigma(a) \wedge \sigma(b)) h\left(\mathscr{N}^{-}(a) \cap \mathscr{N}^{-}(b)\right)$.

数学代写|图论代写GRAPH THEORY代考|pCompetition Fuzzy Graph

FCompG 的另一种变体，类似于模胗 $k$ 竞争图是 $p$ 本节讨论的竞争模湖图。在 $k$-竞赛 $\mathrm{FG}, k$ 是一个实 数，但在这里 $p$ 是一个正整数。所以， $p$ ‘ 是’ 的限制性安例 $k$ 。

(见图 4.5a) 。设

\begin{aligned} 0.8, \sigma\left(x_3\right) &=0.9, \sigma\left(a_1\right)=0.75, \sigma\left(a_2\right)=0.85, \sigma\left(a_3\right)=0.95 \text { 并且对于边缘是 } \ \stackrel{\longrightarrow}{\longrightarrow} \ \mu \overrightarrow{\left.x_1, a_1\right)} &=0.7, \vec{\mu}\left(\overrightarrow{\left.x_1, a_2\right)}=0.7, \mu \overrightarrow{\left(x_2, a_1\right)}=0.6, \mu \overrightarrow{\left(x_2, a_2\right)}=0.8, \mu \overrightarrow{\left(x_2, a_3\right)}=\right.\end{aligned}
$0.8, \mu \overrightarrow{\left(x_3, a_3\right)}=0.9 . \overline{\left(\overrightarrow{\left.a_2, a_3\right)}\right.}=1 . \mu \overrightarrow{\left(a_2, a_1\right)}=1$. 图 4.5a 的有向图的 2 次竞争 $F G$ 如图 $4.5 \mathrm{~b}$ 所 示。

数学代写图论代写GRAPH THEORY代考|m- Step Fuzz Competition Graph

$\rightarrow$ 假设有向图 $\vec{D}$ 和一个正整数 $m$ 给出。这 $m$ – 步骤有向图 $\vec{D}$ 表示为 $\vec{D}^m$ ，在哪里 $\mathscr{V}\left(\vec{D}^m\right)=\mathscr{V}(\vec{D})$ 并且有一个边缘 $(a, b)$ 在 $\vec{D}^m$ 当且仅当存在一个有向长度的游走 $m$ 从 $a$ 至 $b$ 在 $\vec{D}$.

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。