Multiple Choice Questions
Discrete Mathematics: Graph Theory and Networks
Topic: Graph Theory and Networks
Grade: 11
Question 1:
What is the maximum number of edges in a bipartite graph with n vertices on one side and m vertices on the other side?
a) n+m
b) n*m
c) n^2+m^2
d) n*m/2
Answer: d) n*m/2
Explanation: In a bipartite graph, every vertex on one side is connected to every vertex on the other side. Therefore, the maximum number of edges in a bipartite graph with n vertices on one side and m vertices on the other side is n*m. However, since each edge is counted twice (once for each vertex it is connected to), we divide n*m by 2 to get the actual number of edges. For example, if there are 4 vertices on one side and 3 vertices on the other side, the maximum number of edges would be 4*3/2 = 6.
Question 2:
Which of the following is NOT a planar graph?
a) Complete graph K5
b) Complete bipartite graph K3,3
c) Cycle graph C6
d) Tree
Answer: b) Complete bipartite graph K3,3
Explanation: A planar graph is a graph that can be drawn on a plane without any edges crossing. Both the complete graph K5 and the cycle graph C6 are planar graphs. A tree is also a planar graph since it can be drawn without any edges crossing. However, the complete bipartite graph K3,3 cannot be drawn on a plane without any edges crossing. For example, if we have 3 vertices on one side and 3 vertices on the other side, connecting every vertex on one side to every vertex on the other side would result in edges crossing.
Question 3:
In a connected graph with n vertices and n-1 edges, which of the following statements is NOT true?
a) The graph is a tree
b) The graph is a complete graph
c) The graph is a cycle
d) The graph is bipartite
Answer: b) The graph is a complete graph
Explanation: A connected graph with n vertices and n-1 edges is a tree. A tree is a graph that is connected and has no cycles. It is not necessarily a complete graph, which is a graph where every pair of distinct vertices is connected by an edge. For example, a tree with 4 vertices and 3 edges is not a complete graph because not every pair of vertices is connected by an edge.
Question 4:
What is the minimum number of edges in a connected graph with n vertices and k components?
a) n-k
b) n-k+1
c) n-1
d) n-k+2
Answer: a) n-k
Explanation: In a connected graph, every vertex is connected to at least one other vertex. Therefore, the minimum number of edges in a connected graph with n vertices is n-1. However, if the graph has k components, there must be k-1 additional edges to connect the components together. Therefore, the minimum number of edges in a connected graph with n vertices and k components is n-1+(k-1) = n-k. For example, if we have 6 vertices and 3 components, the minimum number of edges would be 6-3 = 3.
Question 5:
Which of the following statements is true about Eulerian graphs?
a) Every vertex has an even degree
b) Every vertex has an odd degree
c) There is no Eulerian path in the graph
d) There is no Eulerian circuit in the graph
Answer: a) Every vertex has an even degree
Explanation: An Eulerian graph is a graph that has a closed trail that visits every edge exactly once, called an Eulerian circuit. In an Eulerian graph, every vertex has an even degree because each time we enter a vertex, we must leave it, using two edges. For example, in a graph where all vertices have degree 2, we can start at any vertex and keep following the edges until we return to the starting vertex, forming an Eulerian circuit.
Question 6:
A graph is said to be Hamiltonian if:
a) It contains a Hamiltonian path
b) It contains a Hamiltonian cycle
c) It contains a Eulerian path
d) It contains a Eulerian cycle
Answer: b) It contains a Hamiltonian cycle
Explanation: A Hamiltonian cycle is a cycle that visits every vertex exactly once. A graph is said to be Hamiltonian if it contains a Hamiltonian cycle. For example, in a complete graph with 4 vertices, we can start at any vertex and visit the remaining 3 vertices in any order, forming a Hamiltonian cycle.
Question 7:
What is the maximum number of edges in a simple graph with n vertices?
a) n*(n-1)
b) n*(n-1)/2
c) n^2
d) n^2/2
Answer: b) n*(n-1)/2
Explanation: In a simple graph, there can be at most one edge between any two vertices. Therefore, the maximum number of edges in a simple graph with n vertices is the number of ways to choose 2 vertices out of n, which is n*(n-1)/2. For example, if there are 5 vertices, the maximum number of edges would be 5*(5-1)/2 = 10.
Question 8:
Which of the following statements is true about a complete bipartite graph K3,4?
a) It has 7 vertices
b) It has 6 edges
c) It is a planar graph
d) It is a tree
Answer: b) It has 6 edges
Explanation: A complete bipartite graph K3,4 has 3 vertices on one side and 4 vertices on the other side, resulting in a total of 7 vertices. Since every vertex on one side is connected to every vertex on the other side, there are 3*4 = 12 edges in total. However, each edge is counted twice (once for each vertex it is connected to), so the actual number of edges is 12/2 = 6.
Question 9:
In a directed graph, a vertex with no incoming edges is called:
a) Source vertex
b) Sink vertex
c) Isolated vertex
d) In-degree zero vertex
Answer: a) Source vertex
Explanation: In a directed graph, a vertex with no incoming edges is called a source vertex. It is called a source vertex because it is the starting point of a flow of edges. For example, in a directed graph where the edges represent dependencies between tasks, a source vertex would represent a task that does not depend on any other task.
Question 10:
What is the minimum number of edges in a directed graph with n vertices and a directed path from every vertex to every other vertex?
a) n-1
b) n
c) n+1
d) n*(n-1)/2
Answer: d) n*(n-1)/2
Explanation: In a directed graph with a directed path from every vertex to every other vertex, every vertex must be connected to every other vertex. Therefore, the minimum number of edges in such a graph is the number of ways to choose 2 vertices out of n, which is n*(n-1)/2. For example, if there are 4 vertices, the minimum number of edges would be 4*(4-1)/2 = 6.
Question 11:
Which of the following statements is true about a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It is a connected graph
d) It is a complete graph
Answer: a) It has n-1 edges
Explanation: A tree is a connected graph with no cycles. Since every vertex in a tree is connected to exactly one other vertex (except for the root vertex), the number of edges in a tree is always one less than the number of vertices. Therefore, a tree with n vertices has n-1 edges. For example, a tree with 5 vertices would have 5-1 = 4 edges.
Question 12:
What is the maximum number of edges in an acyclic directed graph with n vertices?
a) n-1
b) n
c) n+1
d) n*(n-1)/2
Answer: n*(n-1)/2
Explanation: An acyclic directed graph is a directed graph with no cycles. In such a graph, each vertex can be reached from any other vertex through a directed path. Therefore, the maximum number of edges in an acyclic directed graph with n vertices is the same as the maximum number of edges in an undirected simple graph with n vertices, which is n*(n-1)/2. For example, if there are 6 vertices, the maximum number of edges would be 6*(6-1)/2 = 15.
Question 13:
Which of the following statements is true about a planar graph?
a) It can be drawn on a plane without any edges crossing
b) It has a Hamiltonian cycle
c) It has an Eulerian path
d) It has a complete subgraph
Answer: a) It can be drawn on a plane without any edges crossing
Explanation: A planar graph is a graph that can be drawn on a plane without any edges crossing. This means that we can assign coordinates to the vertices and draw the edges as curves without any intersections. For example, a tree is a planar graph because we can draw it without any edges crossing by arranging the vertices in a line.
Question 14:
What is the maximum number of edges in a simple connected graph with n vertices and a diameter of k?
a) n
b) n-1
c) k
d) k+1
Answer: n-1
Explanation: The diameter of a graph is the maximum distance between any pair of vertices. In a simple connected graph with n vertices, the maximum number of edges is n-1. This is because each vertex is connected to at least one other vertex, and we can form a path from any vertex to any other vertex by following the edges. Therefore, the maximum number of edges is n-1, regardless of the diameter of the graph.
Question 15:
Which of the following is NOT a property of a bipartite graph?
a) It contains no odd cycles
b) It can be colored with two colors
c) It has a Hamiltonian cycle
d) It can be divided into two independent sets
Answer: c) It has a Hamiltonian cycle
Explanation: A bipartite graph is a graph where the vertices can be divided into two independent sets such that there are no edges between vertices within the same set. This means that a bipartite graph cannot contain any odd cycles. It can be colored with two colors by assigning one color to one set of vertices and the other color to the other set of vertices. However, it does not necessarily have a Hamiltonian cycle, which is a cycle that visits every vertex exactly once. For example, a bipartite graph with 4 vertices on one side and 3 vertices on the other side does not have a Hamiltonian cycle.