Dept. Computer Science Networks and Graphs
VU University Amsterdam 03.07.2013
BE SURE THAT YOUR HANDWRITING IS READABLE
Part I
1a Let G denote a simple graph with n > 1 vertices and m edges. For each of the following mathematical statements, (1) translate the statement into common English and (2) tell whether it is true or false.
1. ω(G) = 1 ⇒ ∀u ∈ V (G) : δ(v) > 0 2. ∀u ∈ V (G) : δ(v) > 0 ⇒ ω(G) = 1 3. ∑v∈V (G)δ(v) = 2m
4. ∀H ⊆ G : |E(H)| ≤ m
5. |{(u, v)|∃(u, v) − path}| = 0 ⇒ ω(G) = n
10pt
2 Prove that if two graphs have the same degree sequence, they do not also need to be isomorphic. 6pt
3a Consider the original K¨oningsberg bridge graph, shown below. Modify the graph such that it be- comes Eulerian by (1) deleting two edges, (2) adding two edges, and (3) adding an edge and deleting
an edge. 6pt
3b Apply Fleury’s algorithm to find an Euler tour in the following graph. 6pt
4 Prove by induction that, for a given graph G, ∑w∈V (G)δ(w) = 2 · |E(G)|. 10pt
5a Can a directed tree with n > 1 vertices be strongly connected? Explain your answer. 4pt 5b Prove that if a directed graph with n > 1 vertices is strongly connected, every vertex will have a
degree larger than 1. 4pt
5c Consider a strongly connected directed graph G and its underlying undirected graph G∗. Prove that
λ(G∗) ≥ 2. 4pt
Part II
6a Compute for the following graph all the shortest paths to vertex 1 using Dijkstra’s algorithm. Be sure
to make clear how you came to your answer. 6pt
1 2 3
4 6
5 14 2
9 9
10 11
7 15
6
6b Provide a minimal spanning tree T for the graph from (a). Compared to the tree found in (a), T ’s
weight will be the same or less. Why is that so? 4pt
7a Consider a real-world network G with 1000 vertices and 7500 edges. To which random network
should G be compared? 6pt
7b Assume that the clustering coefficient of G from (a) is equal to 0.05. Is this high? Be sure to explain
your answer. 4pt
8a Consider a Watts-Strogatz graph W S(n, k, p). Explain what the parameters n, k, and p stand for. 3pt 8b What will happen to the average path length in a W S(n, k, p) graph when p increases? And what
about the clustering coefficient? 4pt
8c Prove that for any two vertices u and v in a W S(n, k, p) graph, the length of a shortest path between
uand v is less or equal than n/k. 8pt
9a Give the definition of betweenness centrality and an informal description of what it tries to measure. 6pt 9b Compute the betweenness centrality for vertex 2 in the graph from (6a). Explain your answer. 6pt 9c Consider a tree with n vertices {1, 2, . . . , n} in which vertex 1 is joined to all other vertices. What is
the betweenness centrality of vertex 1? 3pt
Final grade: (1) Add, per part, the total points. (2) Let T denote the total points for the midterm exam (0 ≤ T ≤ 50); D1 the total points for part I; D2 the total points for part II. The final number of points E is equal tomax{T, D1} + D2.
2
