Draw a simple graph G with only vertices of degree 3 and λ(G

Resit Exam Networks & Graphs

VU University Amsterdam, 1 July 2015, 12:00-14:45

(Answers can be given in English or Dutch.)

(The exercises in this exam sum up to 90 points; each student gets 10 points bonus.)

Part I

1. Draw a simple graph G with only vertices of degree 3 and λ(G) = 1 (i.e., G has an

edge cut consisting of a single edge). (8 pts)

2. The complement of a simple graph G has vertex set V (G) and edge set {hu, vi | hu, vi /∈ E(G)}.

A simple graph is self-complementary if it is isomorphic to its complement.

(a) How many edges does a self-complementary graph with four vertices have?

(5 pts) (b) Draw a self-complementary graph with four vertices, and argue that it indeed

is self-complementary. (8 pts)

3. Consider a simple graph G with n vertices and m edges. Can G be Eulerian if

(a) n is even and m is odd? (6 pts)

(b) n is odd and m is even? (6 pts)

4. Find a closed walk of minimal weight in the following weighted graph by applying the algorithm to solve the Chinese postman problem. (12 pts)

u v w

x y

4

5 3

7

1 11

3

Part II

5. Apply the Bellman-Ford algorithm to compute shortest paths in the weighted di-

graph below: (12 pts)

u v

w x

−2

3 2

1 3

4

6. Compute the closeness c_C, vertex centrality c_E and betweenness centrality c_B for each of the vertices in the graph below, where each edge is supposed to have weight

1. (12 pts)

y w v

z x

u

7. Argue that Watts-Strogatz graphs in WS (n, k, 0) have a clustering coefficient of approximately 0.75. (12 pts)

8. Explain in detail how Barab´asi-Albert graphs with tunable clustering are con- structed. (9 pts)