如果你也在 怎样代写图论Graph Theory 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。图论Graph Theory有趣的部分原因在于，图可以用来对某些问题中的情况进行建模。这些问题可以在图表的帮助下进行研究(并可能得到解决)。因此，图形模型在本书中经常出现。然而，图论是数学的一个领域，因此涉及数学思想的研究-概念和它们之间的联系。我们选择包含的主题和结果是因为我们认为它们有趣、重要和/或代表主题。

图论Graph Theory通过熟悉许多过去和现在对图论的发展负责的人，可以增强对图论的欣赏。因此，我们收录了一些关于“图论人士”的有趣评论。因为我们相信这些人是图论故事的一部分，所以我们在文中讨论了他们，而不仅仅是作为脚注。我们常常没有认识到数学是一门有生命的学科。图论是人类创造的，是一门仍在不断发展的学科。

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

数学代写|图论作业代写Graph Theory代考|Graphs and Multigraphs

A graph $G$ is a tuple consisting of a finite set $V$ of vertices and a finite set $E$ of edges where each edge is an unordered pair of vertices. The two vertices associated with an edge $e$ are called the end-vertices of $e$. We often denote by $(u, v)$, an edge between two vertices $u$ and $v$. We also denote the set of vertices of a graph $G$ by $V(G)$ and the set of edges of $G$ by $E(G)$. A vertex of a graph is also called as a node of a graph.
We generally draw a graph $G$ by representing each vertex of $G$ by a point or a small circle and each edge of $G$ by a line segment or a curve between its two endvertices. For example, Fig. 2.1 represents a graph $G$ where $V(G)=\left{v_1, v_2, \ldots, v_{11}\right}$ and $E(G)=\left{e_1, e_2, \ldots, e_{17}\right}$. We often denote the number of vertices of a graph $G$ by $n$ and the number of edges of $G$ by $m$; that is, $n=|V(G)|$ and $m=|E(G)|$. We will use these two notations $n$ and $m$ to denote the number of vertices and the number of edges of a graph unless any confusion arises. Thus $n=11$ and $m=17$ for the graph in Fig. 2.1.

A loop is an edge whose end-vertices are the same. Multiple edges are edges with the same pair of end-vertices. If a graph $G$ does not have any loop or multiple edge, then $G$ is called a simple graph; otherwise, it is called a multigraph. The graph in Fig. 2.1 is a simple graph since it has no loop or multiple edge. On the other hand, the graph in Fig. 2.2 contains a loop $e_5$ and two sets of multiple edges $\left{e_2, e_3, e_4\right}$ and $\left{e_6, e_7\right}$. Hence the graph is a multigraph. In the remainder of the book, when we say a graph, we shall mean a simple graph unless there is any possibility of confusion.
We call a graph a directed graph or digraph if each edge is associated with a direction, as illustrated in Fig. 2.3(a). One can consider a directed edge as a one-way street. We thus can think an undirected graph as a graph where each edge is directed in both directions. We deal with digraphs in Chapter 8. We call a graph a weighted graph if a weight is assigned to each vertex or to each edge. Figure 2.3(b) illustrates an edge-weighted graph where a weight is assigned to each edge.

数学代写|图论作业代写Graph Theory代考|Adjacency, Incidence, and Degree

Let $e=(u, v)$ be an edge of a graph $G$. Then the two vertices $u$ and $v$ are said to be adjacent in $G$, and the edge $e$ is said to be incident to the vertices $u$ and $v$. The vertex $u$ is also called a neighbor of $v$ in $G$ and vice versa. In the graph in Fig.2.1, the vertices $v_1$ and $v_3$ are adjacent; the edge $e_1$ is incident to the vertices $v_1$ and $v_3$. The neighbors of the vertex $v_1$ in $G$ are $v_2, v_3 v_6, v_9$, and $v_{11}$.

The degree of a vertex $v$ in a graph $G$, denoted by $\operatorname{deg}(v)$ or $d(v)$, is the number of edges incident to $v$ in $G$, with each loop at $v$ counted twice. The degree of the vertex $v_1$ in the graph of Fig. 2.1 is 5. Similarly, the degree of the vertex $v_5$ in the graph of Fig. 2.2 is also 5 .

Since the degree of a vertex counts its incident edges, it is obvious that the summation of the degrees of all the vertices in a graph is related to the total number of edges in the graph. In fact the following lemma, popularly known as the “Degree-sum Formula,” indicates that summing up the degrees of each vertex of a graph counts each edge of the graph exactly twice.

Lemma 2.2.1 (Degree-sum Formula) Let $G=(V, E)$ be a graph with $m$ edges. Then $\sum_{v \in V} \operatorname{deg}(v)=2 m$.

Proof Every nonloop edge is incident to exactly two distinct vertices of $G$. On the other hand, every loop edge is counted twice in the degree of its incident vertex in $G$. Thus, every edge, whether it is loop or not, contributes a two to the summation of the degrees of the vertices of $G$.

The above lemma, due to Euler (1736), is an essential tool of graph theory and is sometimes refer to as the “First Theorem of Graph Theory” or the “Handshaking Lemma.” It implies that if some people shake hands, then the total number of hands shaken must be even since each handshake involves exactly two hands. The following corollary is immediate from the degree-sum formula.

图论代考

数学代写|图论作业代写Graph Theory代考|Graphs and Multigraphs

图$G$是一个元组，由有限集$V$的顶点和有限集$E$的边组成，其中每条边是一个无序的顶点对。与一条边$e$相关联的两个顶点称为$e$的端顶点。我们通常用$(u, v)$表示两个顶点$u$和$v$之间的一条边。我们也用$V(G)$表示图的顶点集$G$，用$E(G)$表示图的边集$G$。图的顶点也称为图的节点。
我们通常通过用一个点或一个小圆表示$G$的每个顶点，用两个顶点之间的线段或曲线表示$G$的每个边来绘制图形$G$。例如，图2.1表示一个图形$G$，其中$V(G)=\left{v_1, v_2, \ldots, v_{11}\right}$和$E(G)=\left{e_1, e_2, \ldots, e_{17}\right}$。我们通常用$n$表示一个图的顶点数$G$，用$m$表示$G$的边数;即$n=|V(G)|$和$m=|E(G)|$。我们将使用这两个符号$n$和$m$来表示一个图的顶点数量和边的数量，除非出现任何混淆。因此，图2.1中的图形为$n=11$和$m=17$。

循环是端点相同的边。多条边是指具有相同端点对的边。如果一个图$G$没有任何环路或多条边，则称$G$为简单图;否则，它被称为多重图。图2.1中的图是一个简单的图，因为它没有环路和多条边。另一方面，图2.2中的图包含一个循环$e_5$和两组多边$\left{e_2, e_3, e_4\right}$和$\left{e_6, e_7\right}$。因此这个图是一个多图。在本书的其余部分，当我们说图形时，除非有任何混淆的可能，我们将指一个简单的图形。
我们称图为有向图或有向图，如果每条边都与一个方向相关联，如图2.3(a)所示。我们可以把有向边看作单行道。因此，我们可以把无向图看作是每条边都向两个方向有向的图。我们将在第8章讨论有向图。我们称一个图为加权图，如果一个权重被分配给每个顶点或每个边。图2.3(b)显示了一个边加权图，其中每个边都分配了一个权重。

数学代写|图论作业代写Graph Theory代考|Adjacency, Incidence, and Degree

设$e=(u, v)$为图的一条边$G$。那么两个顶点$u$和$v$在$G$中被称为相邻，并且边$e$被称为与顶点$u$和$v$相关。顶点$u$在$G$中也被称为$v$的邻居，反之亦然。在图2.1的图中，顶点$v_1$和$v_3$相邻;边$e_1$与顶点$v_1$和$v_3$相关。$G$中顶点$v_1$的邻居是$v_2, v_3 v_6, v_9$和$v_{11}$。

图$G$中顶点$v$的度数，用$\operatorname{deg}(v)$或$d(v)$表示，是$G$中关联到$v$的边数，在$v$处的每个循环计数两次。图2.1图中顶点$v_1$的度数为5。同样，图2.2中顶点$v_5$的度数也是5。

由于顶点的度数计算了它的关联边，很明显，图中所有顶点的度数之和与图中边的总数有关。事实上，下面的引理，通常被称为“度数和公式”，表明将图中每个顶点的度数相加，图中的每条边都精确地计算两次。

引理2.2.1(度和公式)设$G=(V, E)$是一个有$m$条边的图。然后$\sum_{v \in V} \operatorname{deg}(v)=2 m$。

证明每个非环边都与$G$的两个不同的顶点相关联。另一方面，在$G$中，每个循环边在其入射顶点的度数上被计数两次。因此，无论是否为循环，每条边对$G$的顶点度数之和的贡献都是2。

上述引理，由于欧拉(1736)，是图论的一个重要工具，有时被称为“图论第一定理”或“握手引理”。这意味着如果有些人握手，那么握手的总数一定是偶数，因为每次握手都需要两只手。下面的推论是由次和公式直接得出的。

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