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# 数学代写|图论代考GRAPH THEORY代写|MATH7331 Some Basic Definitions

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

Before going to the definition of fuzzy tree, some related terms are recalled first:
Let $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$ be a FG of a crisp graph $G^*=(V, E)$. A fuzzy path in $\mathscr{G}$ is a sequence of distinct vertices $a_0, a_1, \ldots, a_n$ such that $\mu\left(a_{i-1}, a_i\right)>0,1 \leq i \leq n$. Note that the underlying graph of fuzzy path is a crisp path. Like crisp cycle, the fuzzy cycle is defined. A fuzzy path becomes a fuzzy cycle if its end vertices $a_0$ and $a_n$ coincide. The strength of a fuzzy path is the minimum of the membership values of all the edges in the path and the corresponding edge (which attains the minimum) is called the weakest edge.

There lies a big question in FG about the connectivity between two edges because the membership values (we can say the degree of existence) of the edges being different, maybe very less. So, a new term called strength of connectedness between two vertices $a$ and $b$ in $\mathscr{G}$ is defined and it is the maximum of the strengths of all paths between $a$ and $b$ and it is denoted by $\operatorname{CONN}_{\mathscr{G}}(a, b)$.

A fuzzy subgraph $H=\left(\mathscr{V}, \sigma^{\prime}, \mu^{\prime}\right)$ is called a partial fuzzy subgraph of $\mathrm{FG}$ $\mathscr{G}$ if $\sigma^{\prime}(a) \leq \sigma(a)$ for all $a \in \mathscr{V}$ and $\mu^{\prime}(a, b) \leq \mu(a, b)$ for all edge $(a, b)$ of $H$.
A fuzzy subgraph $H$ of a $\mathrm{FG} \mathscr{G}=(\mathscr{V}, \sigma, \mu)$ is called a full fuzzy subgraph of $\mathscr{G}$ if $\sigma(a)>0$ for all $a \in \mathscr{V}$ and $\mu(a, b)>0$ for all edge $(a, b)$ of $\mathscr{G}$.
Two vertices $a$ and $b$ in $\mathscr{G}$ are called neighbors if $\mu(a, b)>0$.
Let $\mathscr{G}_1=\left(\mathscr{V}, \sigma_1, \mu_1\right)$ and $\mathscr{G}_2=\left(\mathscr{V}, \sigma_2, \mu_2\right)$ be two fuzzy graphs. The $\mathrm{FG} \mathscr{G}_2$ is said to be spanning subgraph of $\mathscr{G}_1$ if $\sigma_1(a)=\sigma_2(a)$ for all $a \in \mathscr{V}$ and $\mu_2(a, b)<$ $\mu_1(a, b)$ for all $a, b$. Note that the sets of vertices of two FGs are same only the membership values of edges are strictly less than the other.

## 数学代写|图论代写GRAPH THEORY代考|Fuzzy Cut Vertex and Fuzzy Bridge

In a connected crisp graph, a vertex $a$ is called a cut vertex if its removal disconnects the graph. Similarly, an edge $(a, b)$ is called a bridge if its removal disconnects the graph. But, in FG the concepts of fuzzy cut vertex and bridge are different. In FG, the removal of fuzzy cut vertex or bridge reduces the connectedness of the FG .
The fuzzy bridge and fuzzy cut vertex are defined below and are illustrated by examples.

Definition $3.2$ (Fuzzy bridge) An edge $(a, b)$ is said to be a fuzzy bridge of a FG $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$ if the strength of connectedness between a pair of vertices is reduced after the removal of $(a, b)$, i.e. there is at least a pair of vertices $u, v \in \mathscr{V}$ such that $\operatorname{CONN}{\mathscr{G}}(u, v)>\operatorname{CONN}{\mathscr{G}-(a, b)}(u, v)$.

Thus, the edge $(a, b)$ is a bridge if and only if there exists vertices $u$, $v$ such that $(a, b)$ is an edge of every strongest path from $u$ to $v$.

Definition $3.3$ (Fuzzy cut vertex) A vertex $c$ is said to be a fuzzy cut vertex of a FG $\mathscr{G}=(\mathscr{V}, \sigma, \mu)$ if removal of it reduces the strength of connectedness between some $a, b, c$ all are distinct.

Thus, $c$ is a cut vertex if and only if there exist two vertices $a, b \in \mathscr{V}$ other than $c$ such that $c$ is on every strongest path from $a$ to $b$.

Example 3.1 Let us consider the $\mathrm{FG} \mathscr{G}=(\mathscr{V}, \sigma, \mu)$ containing three vertices $a, b, c$ and three edges $(a, b),(b, c)$, and $(c, a)$ with membership values $1,0.5,0.5$ respectively (see Fig. 3.6). In this graph, the edge $(a, b)$ is a fuzzy bridge, since $\operatorname{CONN}{\mathscr{G}}(a, b)=1>\operatorname{CONN}{\mathscr{G}-(a, b)}(a, b)=0.5$.
But, there is no cut vertex in this FG.

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

$\mu\left(a_{i-1}, a_i\right)>0,1 \leq i \leq n$. 请注意，模楜路径的底层图是一条清晰的路径。像清晰循环一样，定义了模楜徉环。一条模胡路径 如果它的末端顶点变成一个模楜环 $a_0$ 和 $a_n$ 重合。模湖路径的强度是路径中所有边的隶属度值的最小值，对应的边（达到最小值） 称为最弱边。

FG 中存在一个关于两条边之间的连通性的大问题，因为边的成员值 (我们可以说存在程度) 不同，可能非常小。因此，一个新术 语称为两个顶点之间的连通性强度 $a$ 和 $b$ 在 $\mathscr{G}$ 是定义的，它是之间所有路径的强度的最大值 $a$ 和 $b$ 它表示为 $\mathrm{CONN} \mathscr{G}(a, b)$.

## 数学代写图论代写GRAPH THEORY代考|Fuzzy Cut Vertex and Fuzzy Bridge

3.6）。在该图中，边 $(a, b)$ 是一座模湖的桥，因为 $\operatorname{CONN} \mathscr{G}(a, b)=1>\operatorname{CONN} \mathscr{G}-(a, b)(a, b)=0.5$.

## MATLAB代写

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