Top MCQs on Graph Theory in Mathematics

Question 1

Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?

• 7
  

• 1
  

• 1/8
  

• 8
  

Question 2

Which of the following statements is/are TRUE for undirected graphs? 

P: Number of odd degree vertices is even.
Q: Sum of degrees of all vertices is even. 

• Neither P nor Q
  

• Both P and Q
  

• Q only
  

• P only
  

Question 3

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 the following statements is/are TRUE?

(P) The line graph of a cycle is a cycle.
(Q) The line graph of a clique is a clique.
(R) The line graph of a planar graph is planar.
(S) The line graph of a tree is a tree. 

• P only
• P and R only
• R only
• P, Q and S only

Question 4

Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to

• 6
  

• 5
  

• 4
  

• 3
  

Question 5

Which of the following graphs is isomorphic to

• A
• B
• C
• D

Question 6

Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to

• 360
  

• 45
  

• 30
  

• 15
  

Question 7

• Q3 is planar while K4 is not
  

• Neither K4 nor Q3 are planar
  

• Both K4 and Q3 are planar
  

• K4 is planar while Q3 is not
  

Question 8

Let G = (V,E) be a graph. Define ξ(G) = Σd id x d, where id is the number of vertices of degree d in G. If S and T are two different trees with ξ(S) = ξ(T),then

• |S| = 2|T|
• |S| = |T|-1
• |S| = |T|
• |S| = |T|+1

Question 9

The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?  

(I) 7, 6, 5, 4, 4, 3, 2, 1
(II) 6, 6, 6, 6, 3, 3, 2, 2
(III) 7, 6, 6, 4, 4, 3, 2, 2
(IV) 8, 7, 7, 6, 4, 2, 1, 1 

• I and II
  

• III and IV
  

• IV only
  

• II and IV
  

Question 10

What is the chromatic number of an n-vertex simple connected graph which does not contain any odd length cycle? Assume n >= 2.

• n-1
  

• 3
  

• 2
  

• n