Discrete Mathematics

Question 1
Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?
MSQ
A
The diagonal entries of A^2 are the degrees of the vertices of the graph.
B
If the graph is connected, then none of the entries of A^{n-1}+I_ncan be zero.
C
If the sum of all the elements of A is at most 2(n-1), then the graph must be acyclic.
D
If there is at least a 1 in each of A's rows and columns, then the graph must be connected.
GATE CSE 2022      Graph Theory
Question 2
Consider the following recurrence:
\begin{aligned} f(1)&=1; \\ f(2n)&=2f(n)-1, & \text{for }n \geq 1; \\ f(2n+1)&=2f(n)+1, & \text{for }n \geq 1. \end{aligned}
Then, which of the following statements is/are TRUE?
MSQ
A
f(2^n-1)=2^n-1
B
f(2^n)=1
C
f(5 \dot 2^n)=2^{n+1}+1
D
f(2^n+1)=2^n+1
GATE CSE 2022      Recurrence
Question 3
The following simple undirected graph is referred to as the Peterson graph.

Which of the following statements is/are TRUE?
MSQ
A
The chromatic number of the graph is 3.
B
The graph has a Hamiltonian path.
C
The following graph is isomorphic to the Peterson graph.

D
The size of the largest independent set of the given graph is 3. (A subset of vertices of a graph form an independent set if no two vertices of the subset are adjacent.)
GATE CSE 2022      Graph Theory
Question 4
Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the trace of
A
A^3
B
A^3 divided by 2
C
A^3 divided by 3
D
A^3 divided by 6
GATE CSE 2022      Graph Theory
Question 5
Which one of the following is the closed form for the generating function of the sequence \{a_n \}_{n \geq 0} defined below?
a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.
A
\frac{x(1+x^2)}{(1-x^2)^2}+\frac{1}{1-x}
B
\frac{x(3-x^2)}{(1-x^2)^2}+\frac{1}{1-x}
C
\frac{2x}{(1-x^2)^2}+\frac{1}{1-x}
D
\frac{x}{(1-x^2)^2}+\frac{1}{1-x}
GATE CSE 2022      Function
Question 6
The number of arrangements of six identical balls in three identical bins is ____.
A
7
B
8
C
12
D
5
GATE CSE 2022      Probability Theory
Question 7
Consider a simple undirected graph of 10 vertices. If the graph is disconnected, then the maximum number of edges it can have is .
A
72
B
36
C
16
D
48
GATE CSE 2022      Graph Theory
Question 8
Which of the following statements is/are TRUE for a group G?
MSQ
A
If for all x,y \in G, (xy)^2=x^2y^2, then G is commutative.
B
If for all x \in G, x^2=1, then G is commutative. Here, 1 is the identity element of G.
C
If the order of G is 2 , then G is commutative.
D
If G is commutative, then a subgroup of G need not be commutative.
GATE CSE 2022      Group Theory
Question 9
In a directed acyclic graph with a source vertex s, the quality-score of a directed path is defined to be the product of the weights of the edges on the path. Further, for a vertex v other than s, the quality-score of v is defined to be the maximum among the quality-scores of all the paths from s to v. The quality-score of s is assumed to be 1.

The sum of the quality-scores of all vertices on the graph shown above is ______
A
929
B
254
C
639
D
879
GATE CSE 2021 SET-2      Graph Theory
Question 10
Let S be a set of consisting of 10 elements. The number of tuples of the form (A,B) such that A and B are subsets of S, and A\subseteq B is _______
A
49049
B
59049
C
3524
D
854
GATE CSE 2021 SET-2      Set Theory


There are 10 questions to complete.

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