离散数学(双语) - 第九章答案(高辉)

9.1:P595

Ex2: a) Simple graph b) Multigraph c) Pseudograph

Ex20:Team four beat the vertices to which there are edges from team four, namely only team three. The other teams —team one, team two, team five and team six —all beat team four.

9.2:P608

Ex4 For the graph in exercise 1 the sum is 12, there are 6 edges; For the pseudograph in exercise 2, the sum is 26, there are 13 edges; For the pseudograph in exercise 3, the sum is 24, there are 12 edges;

Ex8: in this directed multigraph there are 4 vertices and 8 edges. The degrees are deg -(a)=2,deg +(a)=2; deg -(b)=3,deg +(b)=4; deg -(c)=2,deg +(c)=1; deg -(d)=1,deg +(d)=1;

Ex20 f) We take two copies of Q3 and joining corresponding vertices.

Ex22: This graph is bipartite, with bipartition {a,c} and {b,d,e}

9.3:P618

Ex2: the list is as follows

Vertex adjacent vertices

a b,d

b a,d,e

c d,e

d a,b,c

e b,c

Ex8

0101010111011001

000100101⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦

Ex16: Because of the numbers larger than 1, we need multiple edges in this graph.

Ex36: These graphs are not isomorphic. The second has a vertex of degree 4, whereas the first

9.4:P629

Ex2: a) a path of length 4; b) a path of length 4; c) not a path; d) not a path

Ex12 b) Notice that the sequence a,b,c,d,e,f,a provides a path from every vertex to every other vertex , so this graph is strongly connected.

Ex20: The two graphs are isomorphic: u1↔v2, u2↔v1, u3↔v3, u4↔v4, u5↔v8, u6↔v7, u7↔v5,u8↔v6.

Ex24(a) to find the number of paths of length 2, we need to look at A 2, which is

3121221413222130311

30312223141221213⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦

Since the (3,4)th entry is 0, so there are no paths of length 2.

9.5:P643

Ex4: The graph has no Euler circuit, since the degree of vertex c is odd. There is an Euler path between the two vertices of odd degree. One such path is f,a,b,c,d,e,f,b,d,a,e,c

Ex30:

This graph can have no Hamilton circuit because of the cut edge {c,f}.

9.6:P655

Ex2: the length of a shortest path is 7.

Ex12: In theory, we can use Dijkstra ’s algorithm.

a) the path through Chicago is the fastest.

b) The path via New York is the cheapest.

c) The path via Denver (or the path Via Los Angeles) is the fastest.

d) The path via Dallas( or the path via Chicago) is the fastest.

Ex14:Here we simply assign the weight of 1 to each edge.

9.7:P665

Ex5: This is K 3,3，with parts {a,d,f} and {b,c,e}, therefore it is not planar.

Ex20: this graph is not homeomorphic to K 3,3, since by rerouting the edge between a and h we see that it is planar.