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1、专业好文档离散数学图论部分期末复习辅导一、单项选择题1设图G，vV，则下列结论成立的是 ( ) Adeg(v)=2E Bdeg(v)=EC D解 根据握手定理（图中所有结点的度数之和等于边数的两倍）知，答案C成立。答 C2设无向图G的邻接矩阵为，则G的边数为( )A6 B5 C4 D3解 由邻接矩阵的定义知，无向图的邻接矩阵是对称的即当结点vi与vj相邻时，结点vj与vi也相邻，所以连接结点vi与vj的一条边在邻接矩阵的第i行第j列处和第j行第i列处各有一个1，题中给出的邻接矩阵中共有10个1，故有102=5条边。答 B3已知无向图G的邻接矩阵为，则G有（ ）A5点，8边 B6点，7边 C6点
2、，8边 D5点，7边解 由邻接矩阵的定义知，矩阵是5阶方阵，所以图G有5个结点，矩阵元素有14个1，142=7，图G有7条边。答 Dooooabcd图一oe4如图一所示，以下说法正确的是 ( ) A(a, e)是割边B(a, e)是边割集C(a, e) ,(b, c)是边割集D(d, e)是边割集定义3.2.9 设无向图G=为连通图，若有边集E1E，使图G删除了E1的所有边后，所得的子图是不连通图，而删除了E1的任何真子集后，所得的子图仍是连通图，则称E1是G的一个边割集若边割集为单元集e，则称边e为割边（或桥）解 割边首先是一条边，因为答案A中的是边集，不可能是割边，因此答案A是错误的删除答
3、案B或C中的边后，得到的图是还是连通图，因此答案B、C也是错误的在图一中，删去(d, e)边，图就不连通了，所以答案D正确答 D注：如果该题只给出图的结点和边，没有图示，大家也应该会做如：若图G=，其中V= a, b, c, d, e ，E= (a, b), (a, c) , (a, e) , (b, c) , (b, e) , (c, e) , (e, d)，则该图中的割边是什么？5图G如图二所示，以下说法正确的是 ( )oooabcd图二oAa是割点Bb, c是点割集Cb, d是点割集Dc是点割集定义3.2.7 设无向图G=为连通图，若有点集V1V，使图G删除了V1的所有结点后，所得的子图
4、是不连通图，而删除了V1的任何真子集后，所得的子图仍是连通图，则称V1是G的一个点割集若点割集为单元集v，则称结点v为割点解 在图二中，删去结点a或删去结点c或删去结点b和d图还是连通的，所以答案A、C、D是错误的在图二中删除结点b和c，得到的子图是不连通图，而只删除结点b或结点c，得到的子图仍然是连通的，由定义可以知道，b, c是点割集所以答案B是正确的答 Boooabcd图三o6图G如图三所示，以下说法正确的是 ( ) A(a, d)是割边B(a, d)是边割集C(a, d) ,(b, d)是边割集D(b, d)是边割集解 割边首先是一条边，(a, d)是边集，不可能是割边在图三中，删除答
5、案B或D中的边后，得到的图是还是连通图因此答案A、B、D是错误的在图三中，删去(a, d)边和(b, d)边，图就不连通了，而只是删除(a, d)边或(b, d)边，图还是连通的，所以答案C正确7设有向图（a）、（b）、（c）与（d）如图四所示，则下列结论成立的是( )图四A（a）是强连通的 B（b）是强连通的C（c）是强连通的 D（d）是强连通的复习：定义3.2.5 在简单有向图中，若在任何结点偶对中，至少从一个结点到另一个结点可达的，则称图G是单向（侧）连通的；若在任何结点偶对中，两结点对互相可达，则称图G是强连通的；若图G的底图，即在图G中略去边的方向，得到的无向图是连通的，则称图G是弱
6、连通的显然，强连通的一定是单向连通和弱连通的，单向连通的一定是弱连通，但其逆均不真定理3.2.1 一个有向图是强连通的，当且仅当G中有一个回路，其至少包含每个结点一次单侧连通图判别法：若有向图G中存在一条经过每个结点至少一次的路，则G是单侧连通的。答 A（有一条经过每个结点的回路）问：上面的图中，哪个仅为弱连通的？答：图(d)是仅为弱连通的请大家要复习“弱连通”的概念8设完全图K有n个结点(n2)，m条边，当（ ）时，K中存在欧拉回路Am为奇数 Bn为偶数Cn为奇数 Dm为偶数解 完全图K每个结点都是n-1度的，由定理4.1.1的推论知K中存在欧拉回路的条件是n-1是偶数，从而n为奇数。答 C
7、提示：前面提到n阶无向完全图Kn的每个结点的度数是n-1，现在要问：无向完全图Kn的边数是多少？答：n(n1)/29若G是一个汉密尔顿图，则G一定是( )A平面图 B对偶图C欧拉图 D连通图定义4.2.1 给定图G，若存在一条路经过图G的每个结点一次且仅一次，则该路称为汉密尔顿路；若存在一条回路经过图G的每个结点一次且仅一次，则该回路称为汉密尔顿回路；具有汉密尔顿回路的图称为汉密尔顿图由定义可知，汉密尔顿图是连通图 答 D10若G是一个欧拉图，则G一定是( )A平面图 B汉密尔顿图C连通图 D对偶图定义4.1.1给定无孤立结点图G，若存在一条路经过图G的每条边一次且仅一次，则该路称为欧拉路（即
8、，欧拉路中没有重复的边，并且包含了图中的每条边）若存在一条回路经过图G的每条边一次且仅一次，则该回路称为欧拉回路具有欧拉回路的图就称为欧拉图由定义可知，欧拉图是连通图 答 C11设G是连通平面图，有v个结点，e条边，r个面，则r= ( )Aev2 Bve2Cev2 Dev2答 A（定理4.3.2：欧拉公式v-e+r = 2）问：如果连通平面图G有4个结点，7条边，那么图G有几个面？12无向树T有8个结点，则T的边数为( )A6 B7 C8 D9答 B13无向简单图G是棵树，当且仅当( )AG连通且边数比结点数少1BG连通且结点数比边数少1CG的边数比结点数少1DG中没有回路答 A（定理5.1.
9、1（树的等价定义）14已知一棵无向树T中有8个顶点，4度、3度、2度的分支点各一个，T的树叶数为( )A8 B5 C4 D3解 这棵无向树T有7条边，所有结点的度数之和为14，而4度、3度、2度的分支点各一个共3个结点占用了9度，所以剩下的5个结点占用5度，即这5个结点每个都是1度结点，故有5片树叶答 B15设G是有n个结点，m条边的连通图，必须删去G的( )条边，才能确定G的一棵生成树A BC D答 A（n个结点的连通图的生成树有条边，必须删去条边）16设无向图G的邻接矩阵为，则G的边数为( )A1 B6 C7 D14答 C17如图二（下图）所示，以下说法正确的是 ( )Ae是割点 Ba,
10、e是点割集Cb, e是点割集 Dd是点割集图二答 A18设有向图（a）、（b）、（c）与（d）如图六（下图）所示，则下列结论成立的是( )图六A（a）只是弱连通的 B（b）只是弱连通的C（c）只是弱连通的 D（d）只是弱连通的答 D19无向完全图K4是（ ）A欧拉图 B汉密尔顿图 C非平面图 D树答 B20以下结论正确的是( )A无向完全图都是欧拉图B有n个结点n1条边的无向图都是树C无向完全图都是平面图D树的每条边都是割边答 D二、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 解 设G有x条边，则由握手定理，答 152设给定图G(如右由图所示)，则
11、图G的点割集是 解 从图G中删除结点f，得到的子图是不连通图，即结点集f是点割集；从图G中删除结点c和e，得到的子图是不连通图，而只删除c或e，得到的子图仍然是连通的，所以结点集c, e也是点割集而其他结点集都不满足点割集的定义的集合，所以应该填写：f、c, e答 f、c,e提示：若f是图G的割点，则f是图G的点割集，删除f点后图G是连通吗？3设G是一个图，结点集合为V，边集合为E，则G的结点 等于边数的两倍答 的度数之和4无向图G存在欧拉回路，当且仅当G连通且 答 G的结点度数都是偶数（定理4.1.1的推论）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 ，则在G中存在一
12、条汉密尔顿路答 n-1（定理4.2.2）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 答 W |S|（定理4.2.1）7设完全图K有n个结点(n2)，m条边，当 时，K中存在欧拉回路答 n为奇数（同一、8题）8结点数v与边数e满足 关系的无向连通图就是树答 e=v-1（定理5.1.1（树的等价定义）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 条边后使之变成树解 由握手定理（定理3.1.1）知道图G有182=9 条边，又由定理5.1.1中给出的图T为树的等价定义之一是“图
13、T连通且e=v-1”，可以知道图G的生成树有5条边，从而要删去4条边答 410设正则5叉树的树叶数为17，则分支数为i = 答 4（定理5.2.1：(m-1)i=t-1）三、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存在一条欧拉回路解 错误只有当G是连通图且其结点度数均为偶数时，图G才存在一条欧拉回路2如下图所示的图G存在一条欧拉回路解 错误因为图G有两个奇数度（3度）结点，所以不存在欧拉回路注：这是一个汉密尔顿图，但不是欧拉图。可见汉密尔顿图不一定是欧拉图其实，欧拉图也不一定是汉密尔顿图如下图所示，图（1）是欧拉图又是汉密尔顿图，图（2）是欧拉图但
14、不是汉密尔顿图，图（3）不是欧拉图但它是汉密尔顿图，图（4）不是欧拉图也不是汉密尔顿图。3如下图所示的图G不是欧拉图而是汉密尔顿图图G解 正确图G有4个3度结点a，b，d，f，所以图G不是欧拉图图G有汉密尔顿回路abefgdca，所以图G是汉密尔顿图4设G是一个有7个结点16条边的连通简单图，则G为平面图解 错误由定理4.3.3知，若G有v个结点e条边，且v3，则e3v-6但本题中，1637-6不成立5设G是一个连通平面图，且有6个结点11条边，则G有7个面解 正确由欧拉定理，连通平面图G的结点数为v，边数为e，面数为r，则v-e+r=2于是有r=2-v+e=2-6+11=7问：“完全图K6是
15、平面图”是否正确？答 不正确因为完全图K6有6个结点15条边，且1536-6=12，即e 3v-6对K6不成立，所以K6不是平面图四、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的图形为：（2）图G的邻接矩阵为：（3）图G的每个结点的度数为：，（4）由关于补图的定义3.1.9可知，先在图G补充边画出完全图（见下面左图），然后去掉原图的边，可得图G补图（见下面右图）：
16、注意：补图中，如果没有标出结点v3，则是错的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解 （1）G的图形如左下图：（2）G的邻接矩阵为：（3）图G有5个结点，其生成树有4条边，用Kruskal算法求其权最小的生成树T：第1步，取具最小权1的边(a,c)；第2步，取剩余边中具最小权1的边(c,e)；第3步，取剩余边中不与前2条边构
17、成回路的具最小权2的边(a,b)；第4步，取剩余边中不与前3条边构成回路的具最小权3的边(b,d)所求最小生成树T如下图，其权为注意：在用避圈法求最小的生成树的关键是：“取图中权数最小的边，且与前面取到的边不构成圈”，很多学生只注意到取权数最小的边了，而忽略了“不构成圈”的要求。如果题目给出如解(1) 中所示赋权图，要求用Kruskal算法（避圈法）求出该赋权图的最小生成树，大家应该会吧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）图G有6个结点，其生成树有5条边，用Kruskal算法求其权最小的生成树T：第1步，取具最小权1的边；第2步，取剩余边中具最
18、小权2的边；第3步，取剩余边中不与前2条边构成回路的具最小权3的边；第4步，取剩余边中不与前3条边构成回路的具最小权5的边；第5步，取剩余边中不与前4条边构成回路的具最小权7的边所求最小生成树T如右图（2）该最小生成树的权为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解 （Huffman算法）：首先组合2+3，求带权5, 5, 7, 17, 31的最优二叉树；再组合5+5，求带权7, 10, 17, 31的最优二叉树；然后组合7+10，求带权17, 17, 31的最优二叉树；继续组合17+17，求带权31, 34的最优二叉树；最后组合31+3
19、4，得65，这是树根所带的权。可从树根开始往下画，即得所求最优二叉树T如下图：所求最优二叉树T的权为：讲评：作业中最优二叉树往往都能画对了，但计算总权值时可能会把有些权的层数计算错了，导致总权值计算错误，大家一定要细心。注意：这4个计算题的解题方法大家一定要掌握。补充：教材第101页例3 给定如图3.3.3所示有向图，其邻接矩阵以及邻接矩阵的乘积如下：，从上面的矩阵中可以得到一些结论，如：（1）从A2中第1行第3列的为1可知，结点v1与v3之间有一条长度为2的路；（2）从A3中第1行第2列的为2可知，v1与v2之间有2条长度为3的路；（3）从A4中第2行第2列的为4可知，在结点v2有4条长度为
20、4的回路如果改成问题：试求：（1）图G中结点v1与v3之间长度为2的路径条数； 1条（2）图G中v1与v2之间长度为3的路径条数； 2条（3）图G中经过v2的长度为4的回路条数 4条五、证明题证明题同学一般都做不好，原因是对证明题方法没有掌握，也是对一些概念不清楚所造成的。因此，希望大家认真学习教材和老师讲课中的证明方法，并通过作业逐步掌握做证明题的方法。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与它的补图中的奇数度顶点个数相等证明 设，则是由n阶无向完全图的边删去E所得到的所以对于任意结点，u在G和中的度数之和等于u在中的度数由于n是大于等于3的奇数，从而的每个结点都是偶数度的
21、（度），于是若在G中是奇数度结点，则它在中也是奇数度结点故图G与它的补图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中至少要添加条边才能使其成为欧拉图证明 由定理3.1.2知，k必为偶数要使这k个奇数度结点变成偶数度结点，从而使图G变成欧拉图，可在每两个奇数度结点间添加一条边故在图G中至少要添加条边才能使其成为欧拉图Editors note: Judson Jones is a meteorologist, journalist and photographer. He has freelanced with CNN for four years, covering sev
22、ere weather from tornadoes to typhoons. Follow him on Twitter: jnjonesjr (CNN) - I will always wonder what it was like to huddle around a shortwave radio and through the crackling static from space hear the faint beeps of the worlds first satellite - Sputnik. I also missed watching Neil Armstrong st
23、ep foot on the moon and the first space shuttle take off for the stars. Those events were way before my time.As a kid, I was fascinated with what goes on in the sky, and when NASA pulled the plug on the shuttle program I was heartbroken. Yet the privatized space race has renewed my childhood dreams
24、to reach for the stars.As a meteorologist, Ive still seen many important weather and space events, but right now, if you were sitting next to me, youd hear my foot tapping rapidly under my desk. Im anxious for the next one: a space capsule hanging from a crane in the New Mexico desert.Its like the s
25、et for a George Lucas movie floating to the edge of space.You and I will have the chance to watch a man take a leap into an unimaginable free fall from the edge of space - live.The (lack of) air up there Watch man jump from 96,000 feet Tuesday, I sat at work glued to the live stream of the Red Bull
26、Stratos Mission. I watched the balloons positioned at different altitudes in the sky to test the winds, knowing that if they would just line up in a vertical straight line we would be go for launch.I feel this mission was created for me because I am also a journalist and a photographer, but above al
27、l I live for taking a leap of faith - the feeling of pushing the envelope into uncharted territory.The guy who is going to do this, Felix Baumgartner, must have that same feeling, at a level I will never reach. However, it did not stop me from feeling his pain when a gust of swirling wind kicked up
28、and twisted the partially filled balloon that would take him to the upper end of our atmosphere. As soon as the 40-acre balloon, with skin no thicker than a dry cleaning bag, scraped the ground I knew it was over.How claustrophobia almost grounded supersonic skydiverWith each twist, you could see th
29、e wrinkles of disappointment on the face of the current record holder and capcom (capsule communications), Col. Joe Kittinger. He hung his head low in mission control as he told Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunday.The weather pla
30、ys an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to
31、-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause), he can expect a
32、 lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet (22.7 miles/36.53 kilometers). Here, Fearless Felix will unclip. He will roll back the door.Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.Below, the Earth become
33、s the concrete bottom of a swimming pool that he wants to land on, but not too hard. Still, hell be traveling fast, so despite the distance, it will not be like diving into the deep end of a pool. It will be like he is diving into the shallow end.Skydiver preps for the big jumpWhen he jumps, he is e
34、xpected to reach the speed of sound - 690 mph (1,110 kph) - in less than 40 seconds. Like hitting the top of the water, he will begin to slow as he approaches the more dense air closer to Earth. But this will not be enough to stop him completely.If he goes too fast or spins out of control, he has a
35、stabilization parachute that can be deployed to slow him down. His team hopes its not needed. Instead, he plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).In order to deploy this chute successfully, he will have to slow to 172 mph (2
36、77 kph). He will have a reserve parachute that will open automatically if he loses consciousness at mach speeds.Even if everything goes as planned, it wont. Baumgartner still will free fall at a speed that would cause you and me to pass out, and no parachute is guaranteed to work higher than 25,000
37、feet (7,620 meters).It might not be the moon, but Kittinger free fell from 102,800 feet in 1960 - at the dawn of an infamous space race that captured the hearts of many. Baumgartner will attempt to break that record, a feat that boggles the mind. This is one of those monumental moments I will always
38、 remember, because there is no way Id miss this.If we dont do that it will go on and go on. We have to stop it; we need the courage to do it.His comments came hours after Fifa vice-president Jeffrey Webb - also in London for the FAs celebrations - said he wanted to meet Ivory Coast international Tou
39、re to discuss his complaint.CSKA general director Roman Babaev says the matter has been exaggerated by the Ivorian and the British media.Blatter, 77, said: It has been decided by the Fifa congress that it is a nonsense for racism to be dealt with with fines. You can always find money from somebody t
40、o pay them.It is a nonsense to have matches played without spectators because it is against the spirit of football and against the visiting team. It is all nonsense.We can do something better to fight racism and discrimination.This is one of the villains we have today in our game. But it is only wit
41、h harsh sanctions that racism and discrimination can be washed out of football.The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifas anti-racism taskforce, is in London for the Football Associations 150th anniversary celebrations and will attend Citys Premier League match at
42、 Chelsea on Sunday.I am going to be at the match tomorrow and I have asked to meet Yaya Toure, he told BBC Sport.For me its about how he felt and I would like to speak to him first to find out what his experience was.Uefa hasopened disciplinary proceedings against CSKAfor the racist behaviour of the
43、ir fans duringCitys 2-1 win.Michel Platini, president of European footballs governing body, has also ordered an immediate investigation into the referees actions.CSKA said they were surprised and disappointed by Toures complaint. In a statement the Russian side added: We found no racist insults from
44、 fans of CSKA.Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attache
