This set of Data Structure Multiple Choice Questions & Answers (MCQs) focuses 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

View Answer

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

View Answer

Explanation: Total number of values in the matrix is 4*4=16, out of which 6 entries are non zero and 10 entries are zero. The graph is represented as follows:

1 2 3 4 1 0 0 1 0 2 0 0 0 1 3 1 0 0 1 4 0 1 1 0

3. Adjacency matrix of all graphs are symmetric.

a) False

b) True

View Answer

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(E^{2})

c) O(E)

d) O(V^{2})

View Answer

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

View Answer

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)

View Answer

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

View Answer

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

View Answer

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] ]

View Answer

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] ], The total no. of ways in which every vertex can walk to itself using 2 edges is ________

a) 2

b) 4

c) 6

d) 8

View Answer

Explanation: A

^{2}= [ [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

View Answer

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

View Answer

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?

#include <bits/stdc++.h> using namespace std; int cur=0; int G[10][10]; bool visited[10]; deque <int> q; 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]; for(int i=0;i<n;i++) visited[i]=false; fun(n); return 0; } void fun(int n) { cout<<cur<<" "; visited[cur]=true; q.push_back(cur); do { for(int j=0;j<n;j++) { if(G[cur][j]==1 && !visited[j]) { q.push_back(j); cout<<j<<" "; visited[j]=true; } } q.pop_front(); if(!q.empty()) cur=q.front(); }while(!q.empty()); }

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

View Answer

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 won’t run infinitely? The Input would be in the form of an adjacency Matrix and n is its dimension (1<n<10).

#include <bits/stdc++.h> 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) All simple graphs

View Answer

Explanation: For any graph (except empty 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

View Answer

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

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