Multiple Choice Questions on Graph Theory Graphs and Networks
Eulerian and Hamiltonian Paths Graph Coloring
Each question is followed by its correct answer and a detailed solution.
1. 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) 2n
Answer: b) n(n-1)/2
Solution: In a simple graph (no self-loops or multiple edges), each vertex can connect to at most n-1
other vertices. This gives n(n-1) edges, but since each edge is counted twice (once from each end),
we divide by 2. Thus, the maximum number of edges is n(n-1)/2.
2. A graph G has 6 vertices and 15 edges. What is the sum of the degrees of all vertices in G?
a) 15
b) 30
c) 60
d) 90
Answer: b) 30
Solution: By the Handshaking Lemma, the sum of degrees of all vertices in a graph is always twice
the number of edges. Here, number of edges = 15, so sum of degrees = 2 * 15 = 30.
3. What is the minimum number of vertices needed to create a simple graph with 5 vertices of
degree 4 and the rest of degree 5?
a) 9
b) 10
c) 11
d) 12
Answer: c) 11
Solution: Let x be the number of vertices of degree 5. The sum of degrees must be even
(Handshaking Lemma). So:
5*4 + 5x = 20 + 5x must be even.
The smallest even value for 5x is 10, so x = 2.
Total vertices = 5 + 2 = 7.
4. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
. In a simple graph, if the degree sequence is (5, 5, 4, 4, 3, 3), how many edges does the graph have?
a) 10
b) 11
c) 12
d) 13
Answer: c) 12
Solution: Sum of degrees = 5 + 5 + 4 + 4 + 3 + 3 = 24
Number of edges = Sum of degrees / 2 = 24 / 2 = 12
5. Which of the following is true for a tree with n vertices?
a) It has n-1 edges
b) It has n edges
c) It has n+1 edges
d) It has 2n edges
Answer: a) It has n-1 edges
Solution: A tree is a connected acyclic graph. For n vertices to be connected without forming a
cycle, there must be exactly n-1 edges.
6. What is the chromatic number of a complete graph with 5 vertices (K5)?
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
Solution: In a complete graph, every vertex is connected to every other vertex. Therefore, each
vertex must have a different color. For K5, we need 5 colors, one for each vertex.
7. A graph G has 7 vertices and 11 edges. What is the minimum number of edges that must be added
to make G Eulerian?
a) 0
b) 1
c) 2
d) 3
Answer: c) 2
Solution: For a graph to be Eulerian, all vertices must have even degree. Currently, total degree = 2
* 11 = 22. To make all degrees even, we need to add 2 more edges, making the total degree 26.
8. In a simple graph with 8 vertices, what is the maximum number of edges possible if the graph is
bipartite?
a) 16
b) 24
c) 28
d) 32
Answer: b) 24
Solution: In a bipartite graph, vertices are divided into two sets, and edges only connect vertices
from different sets. The maximum occurs when the graph is complete bipartite with 4 vertices on
each side. Number of edges = 4 * 4 = 16.
9. What is the number of spanning trees in a complete graph with 4 vertices (K4)?
a) 4
b) 8
c) 16
d) 32
Answer: c) 16
Solution: By Cayley's formula, the number of spanning trees in a complete graph Kn is n^(n-2). For
K4, this is 4^(4-2) = 4^2 = 16.
10. In a planar graph, if V is the number of vertices, E is the number of edges, and F is the number of
faces, what is the relationship between V, E, and F?
a) V + E = F
b) V - E + F = 2
c) V + E + F = 0
d) V - E - F = 2
Answer: b) V - E + F = 2
Solution: This is Euler's formula for planar graphs. It states that for any planar graph, the number
of vertices minus the number of edges plus the number of faces equals 2.
11. What is the degree of each vertex in a complete graph with n vertices?
a) n
b) n-1
c) n+1
d) 2n
Answer: b) n-1
Solution: In a complete graph, each vertex is connected to every other vertex except itself.
Therefore, the degree of each vertex is n-1.
12. A graph G has 10 vertices and 22 edges. What is the average degree of a vertex in G?
a) 2.2
b) 4.4
c) 5.5
d) 11
Answer: b) 4.4
Solution: Total degree = 2 * number of edges = 2 * 22 = 44
Average degree = Total degree / number of vertices = 44 / 10 = 4.4
13. Which of the following is true for a graph to be Eulerian?
a) All vertices have even degree
b) All vertices have odd degree
c) Exactly two vertices have odd degree
d) Exactly one vertex has odd degree
Answer: a) All vertices have even degree
Solution: For a graph to be Eulerian (have an Eulerian circuit), all vertices must have even degree.
This ensures that for every entry into a vertex, there's an exit.
14. What is the minimum number of colors needed to color the vertices of a cycle graph with 5
vertices so that no two adjacent vertices have the same color?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: A cycle graph with an odd number of vertices requires 3 colors. With 2 colors, we would
eventually reach a point where two adjacent vertices must have the same color. 3 colors are
sufficient and necessary.
15. In a simple graph, what is the maximum number of edges that can be present if the graph has 6
vertices and no triangles?
a) 6
b) 9
c) 12
d) 15
Answer: b) 9
Solution: This is known as Turán's theorem. For a triangle-free graph with n vertices, the maximum
number of edges is floor(n^2/4). Here, floor(6^2/4) = floor(36/4) = 9.
16. What is the chromatic number of a bipartite graph?
a) 1
b) 2
c) 3
d) Depends on the number of vertices
Answer: b) 2
Solution: A bipartite graph can always be colored with 2 colors. One color for all vertices in one
partition, and the other color for all vertices in the other partition.
17. In a tournament graph (a directed graph where each pair of vertices is connected by exactly one
directed edge), what is the minimum number of edges that must be reversed to make the graph
transitive?
a) 0
b) 1
c) 2
d) 3
Answer: a) 0
Solution: A tournament graph is already transitive if and only if it is acyclic. The minimum number
of edges that need to be reversed to make a tournament graph transitive is equal to the minimum
number of edges that need to be removed to make it acyclic, which can be 0 if the graph is already
acyclic.
18. What is the diameter of a complete graph with n vertices?
a) 0
b) 1
c) 2
d) n-1
Answer: b) 1
Solution: In a complete graph, every vertex is directly connected to every other vertex. Therefore,
the maximum distance between any two vertices (the diameter) is 1.
19. In a simple connected planar graph with 8 vertices and 20 edges, how many faces does the graph
have?
a) 12
b) 13
c) 14
d) 15
Answer: c) 14
Solution: We can use Euler's formula: V - E + F = 2
Here, V = 8, E = 20
8 - 20 + F = 2
F = 14
20. What is the maximum number of edges in a bipartite graph with 5 vertices on one side and 4
vertices on the other side?
a) 9
b) 16
c) 20
d) 25
Answer: c) 20
Solution: In a bipartite graph, edges can only exist between vertices from different partitions. The
maximum occurs when every vertex from one partition is connected to every vertex from the other
partition. Here, that's 5 * 4 = 20 edges.
21. In a simple graph G with n vertices, if the minimum degree of any vertex is at least (n-1)/2, what
can we conclude about G?
a) G is connected
b) G is complete
c) G is bipartite
d) G is planar
Answer: a) G is connected
Solution: This is a consequence of Dirac's theorem. If the minimum degree is at least (n-1)/2, then
the sum of degrees of any two non-adjacent vertices is at least n-1, which implies the graph is
connected.
22. What is the independence number of a complete graph with 7 vertices?
a) 1
b) 2
c) 3
d) 7
Answer: a) 1
Solution: The independence number is the size of the largest independent set. In a complete
graph, no two vertices are independent (not connected), so the largest independent set has size 1.
23. In a simple graph with 10 vertices, what is the minimum number of edges required to ensure that
the graph is connected?
a) 9
b) 10
c) 45
d) 90
Answer: a) 9
Solution: The minimum number of edges for a connected graph with n vertices is n-1. This forms a
tree. Here, n = 10, so the minimum number of edges is 10-1 = 9.
24. What is the chromatic index of a cycle graph with 6 vertices?
a) 2
b) 3
c) 4
d) 6
Answer: b) 3
Solution: The chromatic index is the minimum number of colors needed to color the edges so that
no two adjacent edges have the same color. For a cycle graph with an even number of vertices, this is
always 2. For an odd number, it's 3. Here, 6 is even, so the answer is 2.
25. In a simple graph G with n vertices, if every vertex has degree at least (n-1)/2, what can we
conclude about G?
a) G has a Hamilton cycle
b) G is Eulerian
c) G is planar
d) G is bipartite
Answer: a) G has a Hamilton cycle
Solution: This is Dirac's theorem. If every vertex in a simple graph with n ≥ 3 vertices has degree ≥
n/2, then the graph has a Hamilton cycle.
26. What is the Wiener index of a path graph with 5 vertices?
a) 10
b) 15
c) 20
d) 25
Answer: c) 20
Solution: The Wiener index is the sum of distances between all pairs of vertices. In a path graph
with 5 vertices, the distances are:
1 + 2 + 3 + 4 + 1 + 2 + 3 + 1 + 2 + 1 = 20
27. In a simple graph G with 8 vertices and 16 edges, what is the average degree of a vertex?
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
Solution: The sum of degrees = 2 * number of edges = 2 * 16 = 32
Average degree = Sum of degrees / number of vertices = 32 / 8 = 4
28. What is the maximum number of edges in a planar graph with 7 vertices?
a) 12
b) 15
c) 18
d) 21
Answer: b) 15
Solution: For a planar graph with n vertices (n ≥ 3), the maximum number of edges is 3n - 6.
Here, 3 * 7 - 6 = 15
29. In a tournament graph with 5 vertices, what is the minimum possible number of edges in the
longest path?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: In any tournament graph, there is always a Hamiltonian path. For 5 vertices, this path has
4 edges. The question asks for the minimum possible number for the longest path, which is 3.
30. What is the chromatic number of a wheel graph with 7 vertices (including the central vertex)?
a) 2
b) 3
c) 4
d) 5
Answer: c) 4
Solution: A wheel graph with an odd number of outer vertices requires 4 colors. The central vertex
needs one color, and the outer cycle (being odd) needs 3 colors.
31. In a simple graph G with 10 vertices, if the complement of G is disconnected, what is the
maximum number of components the complement can have?
a) 2
b) 3
c) 4
d) 5
Answer: d) 5
Solution: If the complement of G is disconnected, G must be the union of two or more complete
subgraphs. The maximum number of components in the complement occurs when G is the union of
as many complete subgraphs as possible. With 10 vertices, this would be K5 ∪ K5, giving 5
components in the complement.
32. What is the matching number of a complete graph with 7 vertices?
a) 2
b) 3
c) 4
d) 7
Answer: b) 3
Solution: The matching number is the size of the maximum matching. In a complete graph with an
odd number of vertices, the maximum matching leaves one vertex unmatched. So, for 7 vertices, the
matching number is (7-1)/2 = 3.
33. In a simple graph G with 8 vertices, if the minimum degree is 4, what is the maximum number of
components G can have?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
Solution: With minimum degree 4, each component must have at least 5 vertices. The maximum
number of components occurs when we have one component with 5 vertices and another with 3
vertices, both complete subgraphs. Any other arrangement would violate the minimum degree
condition.
34.
