Data Structures Multiple Choice Questions and Answers on Adjacency Matrix

1. The number of elements in the adjacency matrix of a graph having 7 vertices is __________

a) 7
b) 14
c) 36
d) 49

Answer: d

Explanation: There are n*n elements in the adjacency matrix of a graph with n vertices.

2. What would be the number of zeros in the adjacency matrix of the given graph ?

a) 10
b) 6
c) 16
d) 0

Answer: b

Explanation: Total number of values in the matrix is 4*4=16, out of which 6 entries are non zero.

3. Adjacency matrix of all graphs are symmetric.

a) False
b) True

Answer: a

Explanation: Only undirected graphs produce symmetric adjacency matrices.

4. The time complexity to calculate the number of edges in a graph whose information in stored in form of an adjacency matrix is ____________

a) O(V)
b) O(E2)
c) O(E)
d) O(V2)

Answer: d

Explanation: As V entries are 0, a total of V^2-V entries are to be examined.

5. For the adjacency matrix of a directed graph the row sum is the _________ degree and the column sum is the ________ degree.

a) in, out
b) out, in
c) in, total
d) total, out

Answer: b

Explanation: Row number of the matrix represents the tail, while Column number represents the head of the edge.

6. What is the maximum number of possible non zero values in an adjacency matrix of a simple graph with n vertices?

a) (n*(n-1))/2
b) (n*(n+1))/2
c) n*(n-1)
d) n*(n+1)

Answer: c

Explanation: Out of n*n possible values for a simple graph the diagonal values will always be zero.

7. On which of the following statements does the time complexity of checking if an edge exists between two particular vertices is not, depends?

a) Depends on the number of edges
b) Depends on the number of vertices
c) Is independent of both the number of edges and vertices
d) It depends on both the number of edges and vertices

Answer: c

Explanation: To check if there is an edge between to vertices i and j, it is enough to see if the value of A[i][j] is 1 or 0, here A is the adjacency matrix.

8. In the given connected graph G, what is the value of rad(G) and diam(G)?

a) 2, 3
b) 3, 2
c) 2, 2
d) 3, 3

Answer: a

Explanation: Value of eccentricity for vertices A, C is 2 whereas for F, B, D, E it is 3.

9. Which of these adjacency matrices represents a simple graph?

a) [ [1, 0, 0], [0, 1, 0], [0, 1, 1] ].
b) [ [1, 1, 1], [1, 1, 1], [1, 1, 1] ].
c) [ [0, 0, 1], [0, 0, 0], [0, 0, 1] ].
d) [ [0, 0, 1], [1, 0, 1], [1, 0, 0] ].

Answer: d

Explanation: A simple graph must have no-self loops, should be undirected.

10. Given an adjacency matrix A = [ [0, 1, 1], [1, 0, 1], [1, 1, 0] ], how many ways are there in which a vertex can walk to itself using 2 edges.

a) 2
b) 4
c) 6
d) 8

Answer: c

Explanation: A2 = [ [2, 1, 1], [1, 2, 1], [1, 1, 2] ], all the 3 vertices can reach to themselves in 2 ways, hence a total of 3*2, 6 ways.

11. If A[x+3][y+5] represents an adjacency matrix, which of these could be the value of x and y.

a) x=5, y=3
b) x=3, y=5
c) x=3, y=3
d) x=5, y=5

Answer: a

Explanation: All adjacency matrices are square matrices.

12. Two directed graphs(G and H) are isomorphic if and only if A=PBP-1, where P and A are adjacency matrices of G and H respectively.

a) True
b) False

Answer: a

Explanation: This is a property of isomorphic graphs.

13. Given the following program, what will be the 3rd number that’d get printed in the output sequence for the given input?

1. #include  

2. using namespace std; 

3. int cur=0; 

4. int G[10][10]; 

5. bool visited[10]; 

6. deque <int> q; 

8. void fun(int n); 

10. int main()

11. {   

12.  int num=0; 

13.  int n; 

14.  cin>>n; 

16.  for(int i=0;i<n;i++) 

17.        for(int j=0;j<n;j++) 

18.          cin>>G[i][j]; 

20.  for(int i=0;i<n;i++) 

21.         visited[i]=false; 

23.         fun(n); 

24.  return 0; 

25. } 

27. void fun(int n)

28. { 

29.  cout<<cur<<" "; 

30.  visited[cur]=true; 

31.  q.push_back(cur); 

33.  do

34.         { 

35.   for(int j=0;j<n;j++)

36.                 { 

37.       if(G[cur][j]==1 && !visited[j])

38.                     { 

39.           q.push_back(j); 

40.           cout<<j<<" "; 

41.           visited[j]=true; 

42.              } 

44.                  } 

46.   q.pop_front(); 

47.   if(!q.empty()) 

48.   cur=q.front(); 

49.   }while(!q.empty()); 

50. }

Input Sequence:-

9 
0 1 0 0 0 0 0 0 1    
1 0 0 0 0 0 0 0 0 
0 0 0 1 1 1 0 0 1 
0 0 1 0 0 0 0 0 0 
0 0 1 0 0 0 0 1 0 
0 0 1 0 0 0 1 0 0 
0 0 0 0 0 1 0 1 1 
0 0 0 0 1 0 1 0 0 
1 0 1 0 0 0 1 0 0

a) 2
b) 6
c) 8
d) 4

Answer: c

Explanation: The given code performs the breadth first search routine on the Graph.
The sequence obtained would be 0 1 8 2 6 3 4 5 7.

14. For which type of graph, the given program would run infinitely? The Input would be in the form of an adjacency Matrix and n is its dimension (1

#include  
using namespace std; 
int G[10][10]; 
void fun(int n); 
 
int main()
{
 int num=0; 
 int n; 
 cin>>n; 
 for(int i=0;i<n;i++) 
   for(int j=0;j<n;j++) 
          cin>>G[i][j]; 
     fun(n); 
 return 0; 
}  
 
void fun(int n)
{ 
 for(int i=0;i<n;i++) 
 for(int j=0;j<n;j++) 
 if(G[i][j]==1) 
 j--; 
}

a) All Fully Connected Graphs
b) All Empty Graphs
c) All Bipartite Graphs
d) None of the Mentioned

Answer: b

Explanation: For any graph having edges, the condition G[i][j]==1 would hold true, which would result in an infinite loop.

15. Given the following adjacency matrix of a graph(G) determine the number of components in the G.

  [0 1 1 0 0 0], 
   [1 0 1 0 0 0],
   [1 1 0 0 0 0],
   [0 0 0 0 1 0],
   [0 0 0 1 0 0],
   [0 0 0 0 0 0].

a) 1
b) 2
c) 3
d) 4

Answer: c

Explanation: 0th 1st and 2nd vertices form a component, 3rd and 4th forms another and 5th vertex forms a component of a single vertex.