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# 数学代写|离散数学代写Discrete Mathematics代考|MAT140 Undirected Graphs, Chains, Cycles, Connectivity

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## 数学代写|离散数学代写Discrete Mathematics代考|Undirected Graphs, Chains, Cycles, Connectivity

The edges of a graph express relationships among its nodes. Sometimes, these relationships are not symmetric, in which case it is desirable to use directed arcs as we have in the previous sections. However, there is a class of problems where these relationships are naturally symmetric or where there is no a priori preferred orientation of the arcs. For example, if $V$ is the population of individuals that were students at Penn between 1900 until now and if we are interested in the relation where two people $A$ and $B$ are related iff they had the same professor in some course, then this relation is clearly symmetric. As a consequence, if we want to find the set of individuals who are related to a given individual $A$, it seems unnatural and, in fact, counterproductive, to model this relation using a directed graph.

As another example suppose we want to investigate the vulnerabilty of an internet network under two kinds of attacks: (1) disabling a node and (2) cutting a link. Again, whether a link between two sites is oriented is irrelevant. What is important is that the two sites are either connected or disconnected.

These examples suggest that we should consider an “unoriented” version of a graph. How should we proceed?

One way to proceed is to still assume that we have a directed graph but to modify certain notions such as paths and circuits to account for the fact that such graphs are really “unoriented.” In particular, we should redefine paths to allow edges to be traversed in the “wrong direction.” Such an approach is possible but slightly awkward and ultimately it is really better to define undirected graphs. However, to show that this approach is feasible, let us give a new definition of a path that corresponds to the notion of path in an undirected graph.

## 数学代写|离散数学代写Discrete Mathematics代考|Trees and Rooted Trees

In this section, until further notice, we are dealing with undirected graphs. Given a graph $G$, edges having the property that their deletion increases the number of connected components of $G$ play an important role and we would like to characterize such edges.

Definition 9.27. Given any graph $G=(V, E, s t)$, any edge $e \in E$, whose deletion increases the number of connected components of $G$ (i.e., $(V, E-{e}, s t\lceil(E-{e})$ ) has more connected components than $G)$ is called a bridge.
For example, the edge $\left(v_{4} v_{5}\right)$ in the graph shown in Figure $9.23$ is a bridge.
Proposition 9.7. Given any graph $G=(V, E, s t)$, adjunction of a new edge e between $u$ and $v$ (this means that st is extended to ste, with $s t_{e}(e)={u, v}$ ) to $G$ has the following effect.

1. Either the number of components of $G$ decreases by 1 , in which case the edge $e$ does not belong to any cycle of $G^{\prime}=\left(V, E \cup{e}, s t_{e}\right)$, or
2. The number of components of $G$ is unchanged, in which case the edge e belongs to some cycle of $G^{\prime}=\left(V, E \cup{e}, s t_{e}\right)$.

# 离散数学代写

## 数学代写|离散数学代写Discrete Mathematics代考|Trees and Rooted Trees

1. 无论是组件的数量 $G$ 椷少 1 ，在这种情况下边烼 $e$ 不属于任何䅕环 $G^{\prime}=\left(V, E \cup e, s t_{e}\right)$ ， 或者
2. 组件的数量 $G$ 不变，在这种情况下，边 $\mathrm{e}$ 属于某个循环 $G^{\prime}=\left(V, E \cup e, s t_{e}\right)$.

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## 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。