Top MCQs on Graph Theory in Mathematics

Question 41

The number of articulation point of the following graph is:

• 0
• 1
• 2
• 3

Question 42

If all the edge weights of an undirected graph are positive, then any subset of edges that connects all the vertices and has minimum total weight is a  

• Hamiltonian cycle
• grid
• hypercube
• tree

Question 43

What is the chromatic number of the following graph?

• 2
• 3
• 4
• 5

Question 44

G is a simple undirected graph. Some vertices of G are of odd degree. Add a node v to G and make it adjacent to each odd degree vertex of G. The resultant graph is sure to be

• regular
• Complete
• Hamiltonian
• Euler

Question 45

Consider the following statements
(I) Let T be a binary search tree with 4 height. The minimum and maximum possible nodes of T are 5 and 15 respectively.
(II) In a binary tree, the number of internal nodes of degree 2 is 6, and the number of internal nodes of degree 1 is 8. The number of leaf nodes in the binary tree is 15.
Which of the following statement(s) is/are correct?

• Only (I)
• Only (II)
• Both (I) and (II)
• Neither (I) nor (II)

Question 46

The line graph L(G) of a simple graph G is defined as follows:

• There is exactly one vertex v(e) in L(G) for each edge e in G.
• For any two edges e and e’ in G, L(G) has an edge between v(e) and v(e’), if and only if e and e’are incident with the same vertex in G.

Which of following option is not correct about "Line Graph"?

• A line graph has an articulation point if and only if the underlying graph has a bridge for which neither endpoint has degree one
• For a graph G with n vertices and m edges, the number of vertices of the line graph L(G) is m, and the number of edges of L(G) is half the sum of the squares of the degrees of the vertices in G, m.
• If a graph G has an Euler cycle, that is, if G is connected and has an even number of edges at each vertex, then the line graph of G is Hamiltonian.
• None of these

Question 47

Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive. Consider the following statements:

1. The path between a pair of vertices in a minimum spanning tree of an undirected graph is necessarily the shortest (minimum weight) path.
2. Minimum Spanning Tree of G is always unique and shortest path between a pair of vertices may not be unique.

Which of the above statements is/are necessarily true?

• (1) only
• (2) only
• both (1) and (2)
• neither (1) nor (2)

Question 48

Consider the following undirected graph G: Choose a value for x that will maximize the number of minimum weight spanning trees (MWSTs) of G. The number of MWSTs of G for this value of x is _________ . Note -This was Numerical Type question.

• 4
• 5
• 2
• 3

Question 49

Let G be a graph with 100! vertices, with each vertex labelled by a distinct permutation of the numbers 1, 2, ..., 100. There is an edge between vertices u and v if and only if the label of u can be obtained by swapping two adjacent numbers in the label of v. Let y denote the degree of a vertex in G, and z denote the number of connected components in G. Then y + 10z = _______ . 

Note -This was Numerical Type question.

• 109

• 110

• 119

• None of these

Question 50

The number of possible min-heaps containing each value from {1, 2, 3, 4, 5, 6, 7} exactly once is _______. Note -This was Numerical Type question.

• 80

• 8

• 20

• 210