Sergio Sanchez - Network Science

In the evolving network model that includes the concept of fitness, how does a node's fitness influence its ability to gain new connections? a) Fitness gives all nodes an equal chance to receive new connections, eliminating the advantage of high-degree nodes. b) Nodes with higher fitness are more likely to receive new connections, even if they currently have a low number of links. c) Fitness reduces the effect of preferential attachment, favoring nodes with fewer connections. d) Fitnes only affects the removal of existing links, not the formation of new ones. e) None of the above. Original Idea by Sergio Sanchez

Given a function f(x), it is known that its first derivative is f'(x) = 3x^2 - 12x + 9. What can be said about the behavior of f(x)? A. The function has a maximum at x = 1. B. The function has a minimum at x = 3. C. The function is increasing for x > 3 and decreasing for x < 1. D. The function is increasing in the intervals (-∞, 1) and (3, ∞), and decreasing in the interval (1, 3). E. None of the above. Original idea by: Sergio Sanchez

Consider the following statements: A. Given a tree T, it is possible to find the diameter of T using two BFS executions. B. If an undirected graph is bipartite, it can be detected using BFS. C. BFS was run on a connected simple graph, starting from a node r. Let u and v be two nodes in the graph. If the distance from r to u is 2, and the distance from r to v is 1, then the distance between u and v in the graph must necessarily be 1. Which of the statements are correct? a) Only A b) Only B c) A and B d) B and C e) None of the above

Let G be a simple, undirected, and connected graph with n vertices. Consider the following statements: A. If (u, v) is a bridge in G, then at least one of the vertices u or v is a cut vertex. B. The maximum number of bridges that G can have is n-1. C. If G is a tree with more than 2 vertices and with f leaves, then the minimum number of edges that need to be added so that there are no bridges is ⌈f/2⌉. Select the alternative that identifies the incorrect statements: a) A and B b) B and C c) Only A d) Only C e) None of the above Original Idea by Sergio Sanchez

During a Depth-First Search (DFS) on a directed graph G, the following edges are observed: The edge (u, v) connects node u to node v, which has already been visited and fully explored. The edge (x, y) connects node x to node y, which has been visited but not fully explored yet. The edge (a, b) connects node a to node b, which has been fully explored, but b belongs to a different branch in the DFS tree. Which of the following options correctly classifies these edges according to their type in a DFS traversal? A) (u, v) is a forward edge, (x, y) is a back edge, (a, b) is a back edge. B) (u, v) is a cross edge, (x, y) is a forward edge, (a, b) is a tree edge. C) (u, v) is a forward or cross edge, (x, y) is a back edge, (a, b) is a cross edge. D) (u, v) is a back edge, (x, y) is a cross edge, (a, b) is a forward edge. E) (u, v) is a cross edge, (x, y) is a back edge, (a, b) is a tree edge. Original Idea by Sergio Sanchez