Graph Theory

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   Discrete Mathematics
Question 2
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   Discrete Mathematics
Question 3
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   Discrete Mathematics
Question 4
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   Discrete Mathematics
Question 5
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   Discrete Mathematics
Question 6
Consider the following directed graph:

Which of the following is/are correct about the graph?
[MSQ]
A
The graph does not have a topological order
B
A depth-first traversal starting at vertex S classifies three directed edges as back edges
C
The graph does not have a strongly connected component
D
For each pair of vertices u and v, there is a directed path from u to v
GATE CSE 2021 SET-2   Discrete Mathematics
Question 7
An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components.
Let T be a DFS tree obtained by doing DFS in a connected undirected graph G.
Which of the following options is/are correct?
A
Root of T can never be an articulation point in G.
B
Root of T is an articulation point in G if and only if it has 2 or more children.
C
A leaf of T can be an articulation point in G.
D
If u is an articulation point in G such that x is an ancestor of u in T and y is a descendent of u in T, then all paths from x to y in G must pass through u.
GATE CSE 2021 SET-1   Discrete Mathematics
Question 8
G is an undirected graph with vertex set {v1, v2, v3, v4, v5, v6, v7} and edge set {v1v2, v1v3, v1v4 ,v2v4, v2v5, v3v4, v4v5, v4v6, v5v6, v6v7 }. A breadth first search of the graph is performed with v1 as the root node. Which of the following is a tree edge?
A
v2v4
B
v1v4
C
v4v5
D
v3v4
ISRO CSE 2020   Discrete Mathematics
Question 9
Graph G is obtained by adding vertex s to K_{3,4} and making s adjacent to every vertex of K_{3,4}. The minimum number of colours required to edge-colour G is _______
A
4
B
5
C
6
D
7
GATE CSE 2020   Discrete Mathematics
Question 10
Let G be an undirected complete graph on n vertices, where n\gt2. Then, the number of different Hamiltonian cycles in G is equal to
A
n!
B
(n-1)!
C
1
D
\frac{(n-1)!}{2}
GATE CSE 2019   Discrete Mathematics


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