Dept. Computer Science Graphs and Networks
VU University Amsterdam 10.02.2009
BE SURE THAT YOUR HANDWRITING IS READABLE
Part I (excludes 5 points bonus)
1 Let G be a simple graph with k components, with each component a tree (i.e., G is a forest). Prove that n = m + k, where n = |V (G)| and m = |E(G)|. You may assume that the statment holds for k = 1. 5 Of course, you can prove this by induction on k. As k= 1 is indeed true (as shown during class), let’s assume that the statement holds for k> 1. Consider a forest with k + 1 components. Simply connect two arbitrarily chosen components by means of an edge, reducing the total number of components to k again. In that case, we now know that n= (m + 1) + k = m + (k + 1), completing the proof.
2a Let G be a simple, disconnected graph. Show that the complement ¯Gof G is connected. 8 Consider two vertices u and v that belong to different components of G, and are thus adjacent in ¯G.
These two nodes cannot have a common neighbor in G, meaning that every vertex in ¯G is adjacent to either u or v. This also means that every two vertices in ¯G are connected through a path containing edgehu, vi.
2b Show by example that if G is a simple, connected graph, that ¯Gcan also be connected. 4
3 Which of the following pairs of graphs are isomorphic? Explain your answer! 6
G1 G2 G3
G1 and G2 are isomorphic, however, when trying to draw G3 in the same “structure” as G2, you will notice that you will fail. This can be shown by drawing the attempt. Alternatively, you can also show that G3 is not bipartite by trying to construct a 2-coloring.
4 Consider the complete bipartite graph Km,n, with m, n > 0. For which values of m and n is Km,n
Eulerian? 4
We know that a graph is Eulerian if and only if it has no vertices with odd degree. This is possible only if both m and n are even.
5 Draw a simple, connected graph with degree sequence (5, 5, 4, 4, 3, 3), and one with degree sequence
(5, 5, 2, 2, 2, 2, 2, 2). 4
6 Give an algorithm that determines whether a directed graph is strongly connected. 8 Simply execute a reachability algorithm for each vertex, such as Algorithm 3.1. We then need to check whether the set of reachable vertices from u is equal to V(D), for every vertex u.
7 Show that any nontrivial connected graph has at least two vertices that are not cut vertices. 6 Let u and v be two vertices at maximal distance d(u, v), where distance is measured in the number of edges of the shortest path. Assume u is a cut vertex and consider G− u . Let w be in the com- ponent not containing v. Because u is assumed to be a cut vertex, there must be a(v, w)-path in G passing through u, meaning that d(v, w) > d(u, v), which contradicts our assumption that d(u, v) was maximal. The same reasoning shows that v can also not be a cut vertex.
Part II (excludes 5 points bonus)
8 Give a (formal or informal) definition of betweenness centrality, and explain what it actually mea-
sures. 4
The betweenness centrality cB(u) of a vertex u is defined as the ratio between the number of shortest paths that pass through u, and the total number of shortest paths:
cB(u) =
∑
x6=y
|S(x, u, y)|
|S(x, y)|
where S(x, u, y) is the set of shortest paths between x and y that pass through u, and S(x, y) the set of shortest paths between x and y. It measures the extent that the distance between any two vertices would change if we would remove vertex u.
9a Give the definition of an Erd¨os-R´enyi random graph ER(n, p). 4
An ER(n, p) graph is a graph with n vertices in which two dstinct vertices are connected by an edge with probability p.
9b Explain that the probability P[δ(v) = k] in an ER(n, p) graph is equal to n−1k pk(1 − p)n−1−k 8 There are a maximum of n− 1 other vertices that can be a neighbor of u. As there are n−1k possi- bilities for choosing k different vertices to be adjacent to u, the probability of having any specific set of k edges connecting u is as given.
9c Compute the clustering coefficient for an ER(n, p) graph. 6
Every vertex with k neighbors can expect a total of k2 · p edges between those neighbors. Because the maximal number of edges between neighbors is k2, the clustering coefficient for each vertex is exactly p.
10a The macroscopic structure of the Web looks like a bowtie. Explain what IN, SCC and OU T stand
for. 6
2
IN stands for the collection of Web pages that have a directed path to pages in SCC, but which cannot be reached from any page in SCC. SCC is the collection of Web pages that form a strongly connected component. OU T is the collection of pages reachable from SCC, but from which the SCC cannot be reached.
10b The Web graph is said to form a scale-free network. Explain what is meant by a scale-free network. 6 This means that the probability that a vertex has degree k is proportional to(1/k)α, with2 < α < 3.
11 Show that the following signed graph is balanced. Explain your answer! 6
-
- +
+ + +
- -
1 2
3
4 5
6
You need to show that we can partition V(G) into two subsets V1and V2such that all positive-signed edges are contained in G[V1] or G[V2], and that all negative-signed edges are between these two sets.
In this case, V1= {1, 2, 3, 4} and V2= {5, 6}.
12 In his articles, Brian Hayes explains how one can model telephone calls as a so-called call graph.
Explain his model and mention one characteristic feature when applying it in practice. 5 Each telephone call consists of two telephones A and B and the fact that A calls B. By simply observing call setups over a period of time, we can get a directed multigraph in which each vertex represents a telephone and A→ B the fact that a call was set up by A to B. This model is simplified by collapsing muliple edges to one, and by discarding directions. In practice, researchers found that call graphs again had a giant component (and, in fact, that they were scale-free).
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.
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