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# 数学代写|组合学代写Combinatorics代考|MAT21018 Regular Graphs

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## 数学代写|组合学代写Combinatorics代考|Regular Graphs

A regular graph is a graph whose vertices all have the same degree. A regular graph with vertices of degree $k$ is called a $k$-regular graph.

Example 8.8.1. Let $G$ be a complete graph with $n$ vertices. The degree of any vertex is $n-1$, and hence $G$ is a regular graph. $\triangle$

Example 8.8.2. All five Platonic graphs are regular. Combinatorial data for the Platonic graphs are given in Table 8.2.1, and we conclude that: a tetrahedron is a 3-regular graph; an octahedron is a 4-regular graph; a hexahedron is a 3-regular graph; an icosahedron is a 5-regular graph; a dodecahedron is a 3 -regular graph. $\triangle$

Example 8.8.3. Graph of n-permutations. Let $G=(V, E)$ be a graph defined as follows. The set of vertices $V$ is the set of all permutations of the set $\mathbb{N}{n}={1,2, \ldots, n}$. Two vertices $v{1}=i_{1} i_{2} \ldots i_{n}$ and $v_{2}=j_{1} j_{2} \ldots j_{n}$ are connected by an edge, if and only if the permutation $v_{2}$ can be obtained from the permutation $v_{1}$ by the transposition of two elements.

The number of vertices of graph $G$ is $|V|=n$ !. The number of edges is $|E|=\frac{n !}{2}\left(\begin{array}{l}n \ 2\end{array}\right)$. All vertices of graph $G$ have the same degree $\left(\begin{array}{l}n \ 2\end{array}\right)$. Hence, graph $G$ is regular. $\triangle$

A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every nonadjacent pair of vertices has the same number of neighbors in common.

## 数学代写|组合学代写Combinatorics代考|Bipartite Graphs

A bipartite graph is a graph $G$ whose vertices can be divided into two disjoint nonempty sets $V_{1}$ and $V_{2}$, such that every edge connects a vertex in $V_{1}$ to one in $V_{2}$. The sets $V_{1}$ and $V_{2}$ are called the sides of bipartite graph $G$. If a bipartite graph is not connected, it may have more than one bipartition.
Example 8.9.1. A bipartite graph with the sides $V_{1}={1,2,3,4,5}$ and $V_{2}={A, B, C, D, E}$ is given in Figure 8.9.1. A possible interpretation of the graph is the following. The vertices from set $V_{1}$ represent five workers, and the vertices from set $V_{2}$ represent five possible positions. An edge that connects a vertex $x \in V_{1}$ and a vertex $X \in V_{2}$ means that the worker $x$ is qualified for the position $X$. $\triangle$

Let $G$ be a bipartite graph. If the two subsets (sides) $V_{1}$ and $V_{2}$ have equal cardinality, then graph $G$ is called a balanced bipartite graph. If all vertices on the same side of the bipartition have the same degree, then $G$ is called biregular. A simple characterization of bipartite graphs is given by the following theorem.

## 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。