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。
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数学代写|图论代写GRAPH THEORY代考|Properties of a Tree
There are many different sets of properties of trees; any of these sets of properties can be taken as a definition of a tree. In this section, we discuss these properties of a tree. We first observe the following two trivial properties of a tree which play crucial roles while dealing with trees.
Lemma 4.2.1 Every tree with two or more vertices has at least two leaves.
Proof Let $T$ be a tree of two or more vertices and let $P$ be a maximal path in $T$. Then the end vertices $u$ and $v$ of $P$ have degree 1 (see Fig.4.1(f)), otherwise $P$ would not be a maximal path in $T$.
Lemma 4.2.2 Every edge in a tree is a cut edge.
Proof Immediate from Lemma 3.1.4.
The following lemma gives some characterization of trees.
Lemma 4.2.3 Let $G$ be a graph with $n$ vertices. Then, any two of the following three statements imply the third (and characterize a tree of $n$ vertices).
(a) $G$ is connected.
(b) $G$ contains no cycle.
(c) $G$ has $n-1$ edges.
Proof (a) \& (b) $\Rightarrow$ (c). We first prove that a connected and acyclic graph $G$ with $n$ vertices has $n-1$ edges. The claim is obvious for $n=1$ since a graph with a single vertex and no cycle has no edges. We thus assume that $n>1$ and the claim is true for any connected and acyclic graph with less than $n$ vertices. We now show that the graph $G$ with $n$ vertices has $n-1$ edges. Since $G$ contains no cycle, every edge $e$ of $G$ is a cut edge. Let $H_1$ and $H_2$ be the two connected components of $G-e$ with $n_1$ and $n_2$ vertices, respectively, where $n_1+n_2=n$. Since both $H_1$ and $H_2$ are acyclic and connected, they contain $n_1-1$ and $n_2-1$ edges respectively. Then the total number of edges in $G$ is $n_1-1+n_2-1+1=n-1$.
数学代写|图论代写GRAPH THEORY代考|Rooted Trees
A rooted tree is a tree in which one of the vertices is distinguished from the others. The distinguished vertex is called the root of the tree. The root of a tree is usually drawn at the top. In Fig. 4.3, the root is $v_1$. If a rooted tree is regarded as a directed graph in which each edge is directed from top to bottom, then every vertex $u$ other than the root is connected by an edge from some other vertex $p$, called the parent of $u$. We also call $u$ a child of vertex $p$. We draw the parent of a vertex above that vertex. For example, in Fig. 4.3, $v_1$ is the parent of $v_2, v_3$, and $v_4$, while $v_2$ is the parent of $v_5$ and $v_6 ; v_2, v_3$, and $v_4$ are the children of $v_1$, while $v_5$ and $v_6$ are the children of $v_2$. A leaf is a vertex of a tree that has no children. An internal vertex is a vertex that has one or more children. Thus every vertex of a tree is either a leaf or an internal vertex. In Fig. 4.3, the leaves are $v_4, v_5, v_6$, and $v_7$, and the vertices $v_1, v_2$, and $v_3$ are internal vertices.
The parent-child relationship can be extended naturally to ancestors and descendants. Suppose that $u_1, u_2, \ldots, u_l$ is a sequence of vertices in a tree such that $u_1$ is the parent of $u_2$, which is a parent of $u_3$, and so on. Then vertex $u_1$ is called an ancestor of $u_l$ and vertex $u_l$ a descendant of $u_1$. The root is ancestor of every other vertex in a tree and every other vertex is a descendant of the root. In Fig. 4.3, $v_1$ is an ancestor of all other vertices, and all other vertices are descendants of the root $v_1$. Note that the definition of ancestor (descendant) does not allow a vertex to be an ancestor (descendant) of itself. However, there are some definitions of ancestor (descendant) which allow a vertex to be an ancestor (descendant) of itself.
A rooted tree is called a binary tree if each vertex has at most two children. In general, a rooted tree is called a $k$-ary tree if each vertex has at most $k$ children. That is, the maximum number of children of a vertex in a $k$-ary tree is $k$.
An ordered rooted tree is a rooted tree in which the children of a vertex is somehow ordered. For example, in a ordered rooted binary tree the children of a vertex are ordered as the left child and the right child. The children of a vertex in a ordered rooted tree may also be ordered in a clockwise or in a counterclockwise order.
图论代写
数学代写|图论代写GRAPH THEORY代考|Properties of a Tree
树有很多不同的属性;这些属性集合中的任何一个都可以作为树的定义。在本节中,我们将讨论树的这些属性。我们首先观察到树的以下两个微不足道的性质,它们在处理树时起着至关重要的作用。
引理4.2.1每个有两个或两个以上顶点的树至少有两个叶子。
证明设$T$是一个有两个或两个以上顶点的树,设$P$是$T$中的一个最大路径。那么$P$的端点$u$和$v$的阶为1(见图4.1(f)),否则$P$就不是$T$中的最大路径。
引理4.2.2树的每条边都是一条切边。
引理3.1.4的直接证明。
下面的引理给出了树的一些特征。
引理4.2.3设$G$是一个有$n$顶点的图。然后,下面三个语句中的任意两个都暗示了第三个(并描述了一个有$n$个顶点的树)。
(a)连接$G$。
(b) $G$不包含循环。
(c) $G$有$n-1$条边。
证明(a) \& (b) \Rightarrow$ (c)。我们首先证明一个有$n$顶点的连通无环图$G$有$n-1$条边。这个结论对于$n=1$是很明显的,因为一个只有一个顶点且没有循环的图没有边。因此,我们假设$n>1$,并且对于任何顶点少于$n$的连通无环图都成立。我们现在证明了有n个顶点的图G有n-1条边。由于$G$不包含循环,因此$G$的每条边$e$都是切边。设$H_1$和$H_2$为$G-e$的两个连通分量,分别具有$n_1$和$n_2$顶点,其中$n_1+n_2=n$。由于$H_1$和$H_2$都是无环且连通的,它们分别包含$n_1-1$和$n_2-1$边。则$G$中的边总数为$n_1-1+n_2-1+1=n-1$。
数学代写|图论代写GRAPH THEORY代考|Rooted Trees
有根树是这样一种树,其中一个顶点与其他顶点是不同的。被区分的顶点称为树的根。树的根通常画在顶部。图4.3中,根为$v_1$。如果把一棵有根树看作是一个有向图,其中的每条边都是从上到下的,那么除了根之外的每个顶点$u$都被另一个顶点$p$的边连接起来,这个顶点$p$被称为$u$的父顶点。我们也称$u$为顶点$p$的子结点。我们在顶点上方画一个顶点的父顶点。例如,在图4.3中,$v_1$是$v_2、v_3$和$v_4$的父结点,$v_2$是$v_5$和$v_6的父结点;V_2、v_3$和v_4$是$v_1$的子节点,$v_5$和$v_6$是$ V_2 $的子节点。叶子是没有子结点的树的顶点。内部顶点是具有一个或多个子顶点的顶点。因此,树的每个顶点要么是叶子顶点,要么是内部顶点。在图4.3中,叶子为$v_4, v_5, v_6$和$v_7$,顶点$v_1, v_2$和$v_3$为内部顶点。
亲子关系可以自然地延伸到祖先和后代。假设$u_1, u_2, \ldots, u_l$是树中的一个顶点序列,使得$u_1$是$u_2$的父结点,而$u_2$又是$u_3$的父结点,以此类推。则顶点$u_1$称为$u_l$的祖先,顶点$u_l$称为$u_1$的后代。根是树中每个其他顶点的祖先,而每个其他顶点是根的后代。在图4.3中,$v_1$是所有其他顶点的祖先,所有其他顶点都是根$v_1$的后代。注意,祖先(后代)的定义不允许一个顶点是它自己的祖先(后代)。然而,有一些祖先(后代)的定义允许顶点成为自身的祖先(后代)。
如果一棵有根树的每个顶点最多有两个子结点,就称为二叉树。一般来说,如果每个顶点最多有$k$个子结点,则根树称为$k$-ary树。也就是说,$k$任意树中顶点的子结点的最大数目是$k$。
有序根树是一种根树,其中顶点的子结点在某种程度上是有序的。例如,在有序根二叉树中,顶点的子节点依次为左子节点和右子节点。在有序根树中,顶点的子结点也可以按顺时针或逆时针顺序排列。
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