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# 数学代写|图论代考GRAPH THEORY代写|MATH913 Definitions and Basic Properties

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## 数学代写|图论代写GRAPH THEORY代考|Definitions and Basic Properties

The IFS defined by Atanasov [3] is cited below.
Definition 9.1 An intuitionistic fuzzy set $A$ defined on the universal set $U$ is characterized as follows $A=\left{\left(x, \mu_A(x), v_A(x)\right): x \in U\right}$, where the membership function $\mu_A: U \rightarrow[0,1]$ and non-membership function $v_A: U \rightarrow[0,1]$ satisfy the condition $0 \leq \mu_A(x)+v_A(x) \leq 1$, for all $x \in U$.

Definition 9.2 The support of an IFS $A=\left(U, \mu_A, v_A\right)$ is defined as $\operatorname{Supp}(A)=$ $\left{x \in U: \mu_A(x) \geq 0\right.$ and $\left.v_A(x) \leq 1\right}$. Also, the support length is $S L(A)=$ $|\operatorname{Supp}(A)|$

Definition 9.3 The core of an IFS $A=\left(U, \mu_A, v_A\right)$ is defined as Core $(A)={x \in$ $U: \mu_A(x)=1$ and $\left.v_A(x)=0\right}$. Also, the core length is $C L(A)=|\operatorname{Core}(A)|$.
Now, we define the height of an IFS
Definition 9.4 The height of an IFS $A=\left(U, \mu_A, v_A\right)$ is defined as
$$\left.h(A)=\sup {x \in U} \mu_A(x), \inf {x \in U} v_A(x)\right)=\left(h_1(A), h_2(A)\right) .$$

Based on the IFS, the IFG is defined. Throughout this chapter, we assume that $G^*=(V, E)$ is the underlying crisp graph of all IFGs.

Definition 9.5 Let $G^=(V, E)$ be a crisp graph. An IFG is an algebraic structure $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$ whose underlying graph is $G^ . \sigma$ and $\mu$ have two components, viz. $\sigma=\left(\sigma_1, \sigma_2\right), \mu=\left(\mu_1, \mu_2\right)$ and
(i) $\sigma_1: \mathscr{V} \rightarrow[0,1]$ and $\sigma_2: \mathscr{V} \rightarrow[0,1]$, denote the degree of membership and nonmembership of the node $a \in \mathscr{V}$ respectively with $0 \leq \sigma_1(a)+\sigma_2(a) \leq 1$ for every $a \in \mathscr{V}$,
(ii) $\mu_1: \mathscr{V} \times \mathscr{V} \rightarrow[0,1]$ and $\mu_2: \mathscr{V} \times \mathscr{V} \rightarrow[0,1]$, where $\mu_1(a, b)$ and $\mu_2(a, b)$ denote the degree of membership and non-membership value of the edge $(a, b)$ respectively such that
\begin{aligned} & \mu_1(a, b) \leq \min \left{\sigma_1(a), \sigma_1(b)\right} \text { and } \ \mu_2(a, b) \leq \max \left{\sigma_2(a), \sigma_2(b)\right} \ 0 \leq \mu_1(a, b)+\mu_2(a, b) \leq 1 \text { for all }(a, b) \in E \end{aligned}

## 数学代写|图论代写GRAPH THEORY代考|Cartesian Product on IFGs

Definition 9.7 The Cartesian product of two IFGs, $\mathscr{G}^{\prime}=\left(\mathscr{V}^{\prime}, E^{\prime}, \sigma^{\prime}, \mu^{\prime}\right)$ and $\mathscr{G}^{\prime \prime}=\left(\mathscr{V}^{\prime \prime}, E^{\prime \prime}, \sigma^{\prime \prime}, \mu^{\prime \prime}\right)$ is defined as $\mathscr{G}=\mathscr{G}^{\prime} \times \mathscr{G}^{\prime \prime}=\left(\mathscr{V}, E, \sigma^{\prime} \times \sigma^{\prime \prime}, \mu^{\prime} \times \mu^{\prime \prime}\right)$, where $\mathscr{V}=\mathscr{V}^{\prime} \times \mathscr{V}^{\prime \prime}$ and $E=\left{\left(\left(p_1, q_1\right),\left(p_2, q_2\right)\right) \mid p_1=p_2,\left(q_1, q_2\right) \in E^{\prime \prime}\right.$ or $q_1$ $\left.=q_2,\left(p_1, p_2\right) \in E^{\prime}\right}$ with
\begin{aligned} &\left(\sigma_1^{\prime} \times \sigma_1^{\prime \prime}\right)\left(p_1, q_1\right)=\sigma_1^{\prime}\left(p_1\right) \wedge \sigma_1^{\prime \prime}\left(q_1\right) \ &\left(\sigma_2^{\prime} \times \sigma_2^{\prime \prime}\right)\left(p_1, q_1\right)=\sigma_2^{\prime}\left(p_1\right) \vee \sigma_2^{\prime \prime}\left(q_1\right) \end{aligned}
for all $\left(p_1, q_1\right) \in \mathscr{V}^{\prime} \times \mathscr{V}^{\prime \prime}$ and
$$\left(\mu_1^{\prime} \times \mu_1^{\prime \prime}\right)\left(\left(p_1, q_1\right),\left(p_2, q_2\right)\right)=\left{\begin{array}{l} \sigma_1^{\prime}\left(p_1\right) \wedge \mu_1^{\prime \prime}\left(q_1, q_2\right), \text { if } p_1=p_2,\left(q_1, q_2\right) \in E^{\prime \prime} \ \mu_1^{\prime}\left(p_1, p_2\right) \wedge \sigma_1^{\prime \prime}\left(q_1\right), \text { if } q_1=q_2,\left(p_1, p_2\right) \in E^{\prime} \end{array}\right.$$
$$\left(\mu_2^{\prime} \times \mu_2^{\prime \prime}\right)\left(\left(p_1, q_1\right),\left(p_2, q_2\right)\right)=\left{\begin{array}{l} \sigma_2^{\prime}\left(p_1\right) \vee \mu_2^{\prime \prime}\left(q_1, q_2\right), \text { if } p_1=q_2,\left(q_1, q_2\right) \in E^{\prime \prime} \ \mu_2^{\prime}\left(p_1, p_2\right) \vee \sigma_2^{\prime \prime}\left(q_1\right), \text { if } q_1=q_2,\left(p_1, p_2\right) \in E^{\prime} \end{array}\right.$$
Definition $9.8$ The degree of the node $\left(p_1, q_1\right)$ in $\mathscr{G}^{\prime} \times \mathscr{G}^{\prime \prime}$ is denoted by
\begin{aligned} &\operatorname{deg}{\mathscr{G}{ }^{\prime} \times \mathscr{G} \prime}\left(p_1, q_1\right)=\left(\operatorname{deg}{1 \mathscr{G} \mathscr{G}^{\prime} \times \mathscr{G}^{\prime \prime}}\left(p_1, q_1\right), \operatorname{deg}{2 \mathscr{G})^{\prime} \times \mathscr{G}^{\prime \prime}}\left(p_1, q_1\right)\right) \text { where } \ &=\sum{p_1=p_2,\left(q_1, q_2\right) \in E^{\prime \prime}} \sigma_1^{\prime}\left(p_1\right) \wedge \mu_1^{\prime \prime}\left(q_1, q_2\right)+\sum_{q_1=q_2,\left(p_1, p_2\right) \in E^{\prime}} \mu_1^{\prime}\left(p_1, p_2\right) \wedge \sigma_1^{\prime \prime}\left(q_1\right) \ &\operatorname{deg}{2 \mathscr{G}} \times \mathscr{G}^{\prime \prime}\left(p_1, q_1\right)=\sum{\left(\left(p_1, q_1\right)\left(p_2, q_2\right)\right) \in E}\left(\mu_2^{\prime} \times \mu_2^{\prime \prime}\right)\left(\left(p_1, q_1\right)\left(p_2, q_2\right)\right) \ &=\sum_{p_1=p_2,\left(q_1, q_2\right) \in E^{\prime \prime}} \sigma_2^{\prime}\left(p_1\right) \vee \mu_2^{\prime \prime}\left(q_1, q_2\right)+\sum_{q_1=q_2,\left(p_1, p_2\right) \in E^{\prime}} \mu_2^{\prime}\left(p_1, p_2\right) \vee \sigma_2^{\prime \prime}\left(q_1\right) . \ & \end{aligned}

## 数学代写|图论代写GRAPH THEORY代考|定义和基本属性

$$\left.h(A)=\sup {x\in U}. \mu_A(x), \inf {x\in U} v_A(x)\right)=/left(h_1(A), h_2(A)\right) 。$$

\begin{aligned} &operatorname{deg}{mathscr{G}{ }^{prime}的时候，mathscr{G}。\prime}\left(p_1, q_1\right)=\left(operatorname{deg}{1 mathscr{G}) \mathscr{G}^{prime} 次，\mathscr{G}^{prime}}左(p_1, q_1\right)，\operatorname{deg}{2 \mathscr{G})^{prime} 次，\mathscr{G}^{prime}}左(p_1, q_1\right) \text { where } \ ＆=sum{p_1=p_2,\left(q_1, q_2\right) \in E^{prime \prime}}. \sigma_1^{prime}\left(p_1\right) \wedgemu_1^{prime prime}\left(q_1, q_2\right)+\sum_{q_1=q_2,\left(p_1, p_2\right)in E^{prime}}. \mu_1^{prime}\left(p_1, p_2\right) \wedge\sigma_1^{prime\prime}\left(q_1\right) &\operatorname{deg}{2 \mathscr{G}} \times \mathscr{G}^{prime \prime}\left(p_1, q_1\right)=\sum{left(p_1, q_1\right)\left(p_2, q_2\right）在E}\left(\mu_2^{prime}\times \mu_2^{prime \prime}\right)\left(\left(p_1, q_1\right)\left(p_2, q_2\right)\right) &=sum_{p_1=p_2,\left(q_1, q_2\right) \in E^{prime \prime}}. \sigma_2^{prime}\left(p_1\right) \vee \mu_2^{prime prime}\left(q_1, q_2\right)+\sum_{q_1=q_2,\left(p_1, p_2\right) \in E^{prime}. \mu_2^{prime}\left(p_1, p_2\right) \vee\sigma_2^{prime \prime}\left(q_1\right) 。\ & \end{aligned}

## MATLAB代写

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