2015年电大离散数学形成性考核作业5答案(图论部分)

形成性考核 作业 1 电大 离散数学作业 5 离散数学 图论部分 形成性考核 书面 作业 本课程形成性考核 书面 作业共 3 次，内容 主要分别是集合论部分、图论部分、数理逻辑部分的综合练习，基本上是按照考试的题型（除单项选择题外）安排练习题目，目的是通过综合性书面作业，使同学自己检验学习成果，找出掌握的薄弱知识点，重点复习，争取尽快掌握。 本次形考 书面 作业是第 二 次作业，大家要 认真及时地完成 图论部分的 综合练习 作业 。 要求： 将此作业用 A4 纸打印出来，手工书写答题， 字迹工整，解答题 要 有解答过程 ，要求 2010年 12 月 5 日前完成并上交任课教师（不收电 子稿）。 并在 05 任务界面下方点击“保存”和“交卷”按钮，以便教师评分。 一 、 填空题 1已知图 G 中有 1 个 1 度结点， 2 个 2 度结点， 3 个 3 度结点， 4 个 4 度结点，则 G 的边数是 15 2 设给定图 G(如 右 由图所示 )，则图 G 的点 割 集 是 f 3 设 G 是一个图，结点集合为 V，边集合为 E，则 G 的 结点 度数 之和 等于边数的两倍 4 无向图 G 存在欧拉回路，当且仅当 G 连通且 等于出度 5 设 G=是具有 n 个结点的简单图，若在 G 中每一对结点度数之和大于等于 n-1 ，则在 G 中存在一条汉密尔顿路 6 若图 G=中具有一条汉密尔顿回路，则对于结点集 V 的每个非空子集 S，在 G 中删除 S中 的所有 结点得到的连通分支数为 W，则 S 中结点 数 |S|与 W 满足的关系式为 W(G-V1) V1 7 设完全图 Kn有 n 个结点 (n2)， m 条 边， 当 n 为奇数 时， Kn中存在欧拉回路 8 结点数 v 与边数 e 满足 e=v-1 关系的无向连通图就是树 9 设图 G 是有 6 个结点的连通图，结点的总度数为 18，则可从 G 中删去 4 条边后使之变成树 10 设正则 5 叉树的树叶数为 17，则分支数为 i = 5 二 、判断说明题 （ 判断下列各题，并说明理由 ） 1 如果图 G 是无向图，且其结点度数均为偶数，则图 G 存在一条欧拉回路 (1) 不正确，缺了一个条件，图 G 应该是连通图，可以找出一个反例，比如图 G 是一个有孤立结点的图。 2 如 下 图所示的图 G 存在一条欧拉回路 姓 名： 学 号： 得 分： 教师签名： 形成性考核 作业 2 (2) 不正确，图中有奇数度结点 ，所以 不存在 是欧拉 回路 。 3 如下图所示的 图 G 不 是欧拉图 而是 汉密尔 顿图 解：正确 因为图中结点 a， b， d， f 的度数都为奇数，所以不是欧拉图。 如果我们沿着 (a,d,g,f,e,b,c,a)，这样除起点和终点是 a 外，我们经过每个点一次仅一次，所以存在一条汉密尔顿回路，是汉密尔顿图 4 设 G 是一个有 7 个结点 16 条边的连通图，则 G 为平面图 解： (1) 错误 假设图 G 是连通的平面图，根据定理，结点数 v，边数为 e，应满足 e 小于等于 3v-6，但现在 16小于等于 3*7-6，显示不成立。所以假设错误。 5 设 G 是一个连通平面图， 且 有 6 个结点 11 条边，则 G 有 7 个面 (2) 正确 根据欧拉定理，有 v-e+r=2，边数 v=11，结点数 e=6，代入公式求出面数 r=7 三 、 计算题 1 设 G=， V= v1， v2， v3， v4， v5， E= (v1,v3)， (v2,v3)， (v2,v4)， (v3,v4)， (v3,v5)， (v4,v5) ，试 (1) 给出 G 的图形表示； (2) 写出其邻接矩阵； (3) 求出每个结点的度数； (4) 画出其补图的图形 解： (1) G v1 v5 v2 v3 v4 形成性考核 作业 3 (2) 邻接矩阵为 0110010110110110110000100 (3) v1结点度数为 1， v2结点度数为 2， v3结点度数为 3， v4结点度数为 2， v5结点度数为 2 (4) 补图图形为 2 图 G=，其中 V= a, b, c, d, e， E= (a, b), (a, c), (a, e), (b, d), (b, e), (c, e), (c, d), (d, e) ，对应边的权值依次为 2、 1、 2、 3、 6、 1、 4 及 5，试 （ 1）画出 G 的图形； （ 2）写出 G 的邻接矩阵； （ 3）求出 G 权最小的生成树及其权值 （ 1） G 的图形如下： （ 2） 写出 G 的邻接矩阵 v1 v5 v2 v3 v4 形成性考核 作业 4 （ 3） G 权最小的生成树及其权值 3已知带权图 G 如右图所示 (1) 求图 G 的最小生成树； (2)计算该生成树的权 值 解： (1) 最小生成树为 (2) 该生成树的权值为 (1+2+3+5+7)=18 4设有一组权为 2, 3, 5, 7, 17, 31， 试画出相应的最优二叉树 ， 计算 该 最优二叉树 的 权 1 2 3 5 7 形成性考核 作业 5 权为 2*5+3*5+5*4+7*3+17*2+31=131 四 、 证明 题 1 设 G 是一个 n 阶无向简单图， n 是大于等于 3 的奇数证明图 G 与它的补图 G 中的奇数度顶点个数相等 证明： 设 ,G V E ， ,G V E 则 E 是由 n 阶无向完全图nK的边删去 E 所得到的所以对于任意结点 uV ， u 在 G 和 G 中的度数之和等于 u 在nK中的度数由于 n 是大于等于 3 的奇数，从而nK的每个结点都是偶数度的（ 1 ( 2)n 度），于是若 uV 在 G 中是奇数度结点，则它在 G 中也是奇数度结点故图 G 与它的补图 G 中的奇数度结点个数相等 2 设连通图 G 有 k 个奇数度的结点，证明在图 G 中至少要添加2k条边才能使其成为欧拉图 证明 ：由定理 3.1.2，任何图中 度数为奇数的结点必是偶数，可知 k 是偶数 又根据定理 4.1.1 的推论，图 G 是欧拉图的充分必要条件是图 G 不含奇数度结点因此只要在每对奇数度结点之间各加一条边，使图 G 的所有结点的度数变为偶数，成为欧拉图 故最少要加2k条边到图 G 才能使其成为欧拉图 3 5 2 5 10 7 17 31 17 34 65 形成性考核 作业 6 请您删除一下内容， O( _ )O 谢谢！ 2015 年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄 After earning his spurs in the kitchens of The Westin, The Sheraton, Sens on the Bund, and a sprinkling of other top-notch venues, Simpson Lu fi nally got the chance to become his own boss in November 2010. Sort of. The Shanghai-born chef might not actually own California Pizza Kitchen (CPK) but he is in sole charge of both kitchen and frontof- house at this Sinan Mansionsstalwart. Its certainly a responsibility to be the head chef, and then to have to manage the rest of the restaurant as well, the 31-year-old tells Enjoy Shanghai. In hotels, for example, these jobs are strictly demarcated, so its a great opportunity to learn how a business operates across the board. It was a task that management back in sunny California evidently felt he was ready for, and a vote of confi dence from a company that, to date, has opened 250 outlets in 11 countries. And for added pressure, the Shanghai branch was also CPKs China debut. For sure it was a big step, and unlike all their other Asia operations that are franchises, they decided to manage it directly to begin with, says Simpson. Two years ago a private franchisee took over the lease, but the links to CPK headquarters are still strong, with a mainland-based brand ambassador on hand to ensure the business adheres to its ethos of creating innovative, hearth-baked pizzas, a slice of PR blurb that Simpson insists lives up to the hype. They are very innovative, he says. The problem with most fast food places is that they use the same sauce on every pizza and just change the toppings. Every one of our 16 pizza sauces is a unique recipe that has been formulated to complement the toppings perfectly. The largely local customer base evidently agrees and on Saturday and Sunday, at least, the place is teeming. The kids-eat-for-free policy at weekends is undoubtedly a big draw, as well as is the spacious second-fl oor layout overlooked by a canopy of green from Fuxing Park over the road. The company is also focusing on increasing brand recognition and in recent years has taken part in outside events such as the regular California Week. Still, the sta are honest enough to admit that business could be better； as good, in fact, as in CPKs second outlet in the popular Kerry Parkside shopping mall in Pudong. Sinan Mansions has really struggled to get the number of visitors that were envisaged when it first opened, and it hasnt been easy for any of the tenants here, adds Simpson. Were planning a third outlet in the city in 2015, and we will probably choose a shopping mall again because of the better foot traffic. The tearooms once frequented by Coco Chanel and Marcel Proust are upping sticks and coming to Shanghai, Xu Junqian visits the Parisian outpost with sweet treats. One thing the century-old Parisian tearoom Angelina has shown is that legendary fashion designer Coco Chanel not only had style and glamor but also boasted great taste in food, pastries in particular. One of the most popular tearooms in Paris, Angelina is famous for having once been frequented by celebrities such as Chanel and writer Marcel Proust. Now Angelina has packed up its French ambience, efficient service, and beautiful, comforting desserts and flown them to Shanghai. At the flagship dine-in and take-out space in Shanghai, everything mimics the original tearoom designed from the beginning of the 20th century, in Paris, the height of Belle Epoque. The paintings on the wall, for example, are exactly the same as the one that depicts the landscape of southern France, the hometown of the owner； and the small tables are intentional imitations of the ones that Coco Chanel once sat at every afternoon for hot chocolate. The famous hot chocolate, known as LAfricain, is a luxurious mixture of four types of cocoa beans imported from Africa, blended in Paris and then shipped to Shanghai. Its sinfully sweet, rich and thick as if putting a bar of melting chocolate directly on the tongue and the fresh whipped cream on the side makes a light, but equally gratifying contrast. It is also sold in glass bottles as takeaway. The signature Mont-Blanc chestnut cake consists of three parts: the pureed chestnut on top, the vanilla cream like stuffing, and the meringue as base. Get all three layers in one scoop, not only for the different textures but also various flavors of sweetness. The dessert has maintained its popularity for a century, even in a country like France, perhaps the worlds most competitive place for desserts. A much overlooked pairing, is the Paris-New York choux pastry and N226 chocolate flavored tea. The choux pastry is a mouthful of airy pecan-flavored whipped cream, while the tea, a blend of black teas from China and Ceylon, cocoa and rose petals, offers a more subtle fragrance of flowers and chocolate. Ordering these two items, featuring a muted sweetness, makes it easier for you to fit into your little black dress. Breakfast, brunch, lunch and light supper are also served at the tearoom, a hub of many cultures and takes in a mix of different styles of French cuisines, according to the management team. The semi-cooked foie gras terrine, is seductive and deceptive. Its generously served at the size and shape of a toast, while the actual brioche toast is baked into a curved slice dipped with fig chutney. The flavor, however, is honest: strong, smooth and sublime. And you dont actually need the toast for crunchiness. This is the season for high teas, with dainty cups of fine china and little pastries that appeal to both visual and physical appetites. But there is one high tea with a difference, and Pauline D. Loh finds out just exactly why it is special. Earl Grey tea and macarons are all very well for the crucial recuperative break in-between intensive bouts of holiday season shopping. And for those who prefer savory to sweet, there is still the selection of classic Chinese snacks called dim sum to satisfy and satiate. High tea is a meal to eat with eye and mouth, an in-between indulgence that should be light enough not to spoil dinner, but sufficiently robust to take the edge off the hunger that strikes hours after lunch. The afternoon tea special at Shang-Xi at the Four Seasons Hotel Pudong has just the right elements. It is a pampering meal, with touches of luxury that make the high tea session a treat in itself. Whole baby abalones are braised and then topped on a shortcrust pastry shell, a sort of Chinese version of the Western vol-au-vent, but classier. Even classier is the dim sum staple shrimp dumpling or hargow, upgraded with the addition of slivers of midnight dark truffles. This is a master touch, and chef Simon Choi, who presides unchallenged at Shang-Xi, has scored a winner again. Sweet prawns and aromatic truffles whats not to love? His masterful craftsmanship is exhibited in yet another pastry a sweet pastry that is shaped to look like a walnut, but which you can put straight into the mouth. It crumbles immediately, and the slightly sweet, nutty morsel is so easy to eat youll probably reach straight for another. My favorite is the dessert that goes by the name yangzhi ganlu, or ambrosia 形成性考核 作业 7 from the gods. The hotel calls it chilled mango cream with sago, pomelo and birds nest made with ingredients that resonate with every female soul. It does taste like ambrosia, with the sweet-sour fragrance of the mango forming the first layer of taste and sensation, and the pomelo sacs and sago pearls providing the tactile contrast. The birds