MO412 - Graph Algorithms Exercises

Regarding scale-free networks and degree distributions with exponent \(\gamma\), which of the following statements is FALSE ? A) If \(\gamma B) If \(2 C) If \(\gamma > 3\), the first and second moment converges D) If \(\gamma = 2\), the maximum degree \(k_{max}\) scales linearly with the network size E) None of the above Original idea by: Fernando de Facio Rossetti

Given two networks: Network A with \(N_{A}\)=50 nodes and \(P_{A}\) Network B with \(N_{B}\)=100 nodes and \(P_{B}\) Analyze these statements and find the values of X and Y I. For both networks to share the same average degree, \(P_{A}\) should be approximately (1 decimal) X times \(P_{B}\) II. Given that \(P_{A} = 0.5\) and \(P_{B} = 0.15\), you need to add Y nodes to A so they share the same average number of links A) X=2, Y=5 B) X=2, Y=6 C) X=3, Y=7 D) X=3, Y=4 E) None of the above Original idea by: Fernando de Facio Rossetti

A bipartite graph can also be interpreted as a \(2\)-colorable graph. That means every vertex is assigned one of two possible colors, such that no adjacent vertices share the same color. This can be generalized to any number of colors, so a graph G can be k-colorable if you assign a color \(c \in \{c1,c2,\dots,c_{k}\}\) to every vertex such that no adjacent vertices share the same color. Figures 1 and 2 show an example of a \(2\)-colorable and \(3\)-colorable graph, respectively. Figure 1 \(2\)-colorable graph Figure 2 \(3\)-colorable graph Which of the following statements are true I. Cycles with odd number of vertices cannot be \(2\)-colorable; II. A complete graph with \(N\) vertices is \(N\)-colorable at minimum; III. An empty graph is not \(2\)-colorable; IV. A graph is \( (k+1) \)-colorable if and only if it is \(k\)-colorable. A) I, II B) I, II, III C) I, II, III, IV D) III, IV E) None of the above Origi...