solved-question-paper-questions-graph-theory1 (1).pdf

Solved Question Paper
Questions graph theory
Unit 1

1.c)State and prove the handshaking theorem. [June 2017, 5
marks]
Handshaking problem :
If G is a (p,q) graph withV(G)={V1….Vp} and di=dG(Vi), 1≤i≤p, then

1.d)Define the following symbols : i) δ (G)[June 2017,1
mark]
Minimum vertex degree of a graph is δ (G). It is min{dG(x) :x∈V(G)} .
δ (G) is a non-negative integer.
δ (G)=1
G

2.a)What is meant by complement of a graph ? Find the
complement of the C5 graph (i.e. C5 ). [June 2017, 3
marks]
Complement of a graph : Let graph G=(V,E) be a (p,q) graph. Complement of the graph is a
graph V( ) =V(G) and E( )= {xy : xy ∉ E (G), x, y ∈V (G)}.
C5

2.b) What is a complete graph ? [June 2017, 2 marks]
Complete graph : Graph in which any two vertices are adjacent, i.e. each vertex is joined to
every other vertex by a vertex. A complete graph on n vertices is represented by Kn.

5.c)Define isomorphism. Determine whether the following
pair of graphs are isomorphic : [June 2017, 3 marks]
Let G=(V(G),E(G)) and H=(V(H),E(H)) be two graphs. Let us map a function f:V(G)->V(H).
Then two graphs are said to be isomorphic, if
i) F is one-one and onto, and
ii) xy ∈ E(G) if and only if f(x) f(y) ∈ E(H)
If not they are called non-isomorphic graphs.
G H

To check for isomorphism check the following :
1. Number of vertices
Number of vertices in G=5
Number of vertices in H=5
2. Number of edges
Number of edges in G=5
Number of edges in H=5
3. Degree sequence
Degree sequence of G : <2,2,2,2,2>

Degree of sequence of H : <2,2,2,2,2>
The above shows that degree sequence of two graphs is the same.
f(u1)=v1, f(u2)=v2, f(u3)=v3, f(u4)=v4, f(u5)=v5
From the above checks, we can conclude that the two graphs are isomorphic.

3.c)What do you mean by isomorphic graphs ?[June 2016, 2
marks]

To check if two graphs check for these conditions :
1. Count the number of vertices – must be equal
2. Count the number of edges – must be equal
3. Degree sequence – must be same
4. Number of cycles – must be same
5. Max degree vertex and min degree vertex
6. Peculiarity of adjacent vertices

To check for isomorphism check the following :
1. Number of vertices
Consider the two graphs:

2. Number of edges
3. Degree sequence
Degree of sequence of H : <2,2,2,2,2>
The above shows that degree sequence of two graphs is the same.
From the above checks, we can conclude that the two graphs are isomorphic.

4.a) State Handshaking Theorem.[June 2016,3 marks]

4.b) A non-directed graph G has 8 edges. Find the number
of vertices, if the degree of each vertex in G is 2. [June
2016, 3 marks]
According to the formula,
q=8
Sum of degree of all vertices < = 2 * no. of edges . [According to Handshaking theorem]
Let n be number of vertices in graph.
==> 2*n = 2*8
==> 2n=16
==> n=8

1.b)Prove that the complement of is G.[December 2016,5
marks]
Let graph G=(V,E) be a (p,q) graph. Complement of the graph is a graph V( ) =V(G) and E( )=
{xy : xy ∉ E (G), x, y ∈V (G)}.
From the above definition, we can say that complement of a graph has,
V( )=V(G) and E( )= {xy : xy ∉ E (G), x, y ∈V (G)}.
Complement of is G
(V( ))’=V(G) and (E( ) )’={xy : xy ∉ E ( ), x, y ∈V (G)}=E(G)
Hence proved.
Example :

G
V(G)={A,B,C,D,E} E(G)={AE,AB,BC,CD,DE}
V( )=V(G)={A,B,C,D,E} E( )={AD,AC,BE,BD,EC}
Similarly :Complement of is G.
Hence proved.

1.c)Draw at least 3 non-isomorphic graphs on 4
vertices.[December 2016,5 marks]

1.c)Determine whether the following graphs are isomorphic.
If yes, justify your answer. [December 2016,December
2010,4 marks]
Number of vertices in G= 4
Degree sequence of G : {4,4,2,2}
Degree sequence of H : {3,3,3,3}
Degree sequences of graph G and H are different, therefore the two graphs are non-isomorphic.

1.d)What is an undirected graph ? Prove that an undirected
graph has even number vertices of odd degree.[December
2016, 4 marks]
Undirected graph G is a finite non-empty setV together with set E containing pairs of points of
V. V is called the vertex set and E is the edge set of G. In undirected graph, E(G) will be
symmetric onV(G). If (u,v) is there, then (v,u) will be there.
Any graph can only have an even number of odd vertices.
Consider a (p,q) graph with {x1,x2,…..xt} is a set of odd vertices and {xt+1,…..xp} is a set of
even vertices.
Let dG(xi)=2ci+1 1≤i≤t and dG(xi)=2ri t+1≤i≤p

2.a)Define n-regular graph. Show for which value of n the
following graphs are regular : (i) Kn (ii) Qn [December
2016, 5 marks]
It is a graph in which each vertex has the same degree. It is said to be regular graph degree of
regularity r. G is an r-regular graph where 0≤r≤(p-1).
i) Kn
Kn is a regular graph with n=3.
The degree of each vertex is 2. So, K3 is regular graph.
Kn for n>3 it is (n–1)-regular.

2.c)How many edges does a complete graph of 5 vertices
have ? [December 2016,2 marks]
Number of edges in a complete graph of n vertices = n(n-1)/2
In the above question,
number of vertices ,n =5
Number of edges = (n(n-1))/2
= (5(5-1))/2
=(5*4)/2
=10

3.b)Define a graph and a subgraph. Show that for a subgraph H of
a graph G Δ (H) ≤ Δ (G). [December 2016S, 5 marks]
A graph is a set of the form {(x,f(x)): x is a domain of function f}.
Example :
Let G = (V (G), E (G)) be a graph. A subgraph H of the graph G is a graph, such that every vertex
of H is a vertex of G, and every edge of H is an edge of G also, that is,V (H) ⊆V (G) and E (H) ⊆ E
(G).

3.a)Show that for a subgraph H of a graph G Δ (H) ≤ Δ (G).
[December 2014, December 2011,June 2010,December 2010, 5marks]
Let x ∈V(H) such that dH(x) = H(Δ)
Then, NH(x) ⊆ NG(x) .Thus,
Δ (H)=| NH (x)| ≤ | NG (x)|≤Δ (G)

2.d)Define Graph and Subgraph. Give an example of a subgraph H
of a graph G with δ (G) < δ (H) and Δ (H) ≤ Δ (G).[June 2015,4
marks]
A graph is a set of the form {(x,f(x)): x is a domain of function f}.
Example :
Let G = (V (G), E (G)) be a graph. A subgraph H of the graph G is a graph, such that every vertex
of H is a vertex of G, and every edge of H is an edge of G also, that is,V (H) ⊆V (G) and E (H) ⊆ E
(G).

δ(G)=1<2= δ(H)
Δ(H)=2<3= Δ (G)
Diagram :

1.a)Define regular graph. Find the number of edges of a 4-
regular graph with 6 vertices.[December 2015,3 marks]
Kn is a regular graph with degree of regularity (n-1) when n > 3.
4-regular graph with 6 vertices:
Number of edges =12

3.c)Define isomorphic graph. Give an example of the
same.[December 2015, 2 marks]

Both are (5, 5)-graphs. Degree sequence of both the graphs is <2,2,2,2,2>. Both these graphs
have a copy of C5.Therefore, both these graphs are isomorphic.

4.b)For the following graph G, draw subgraphs 3 (i) G - e
(ii) G - a . [December 2015,3 marks]
Graph G:
i) G-e ii)
a
e

Define : (i) Simple graph (ii) Finite and infinite graph
(iii) Isolated vertex (iv) Subgraph[June 2014, 4 marks]
i) Simple graph :
Undirected graph that has no loops or multiple edges is called a simple graph.When an edge
joins a vertex to itself is called a loop.Two or more edges that joins the same vertices are
parallel or multiple edges.
ii) Finite and infinite graph : A graph with a finite number of vertices and edges is called a finite
graph. A graph with a finite number of nodes and edges.

iii)Isolated vertex :Vertex with degree zero is called an isolated vertex.
iv) Subgraph : Let G = (V (G), E (G)) be a graph. A subgraph H of the graph G is a graph, such
that every vertex of H is a vertex of G, and every edge of H is an edge of G also, that is,V (H) ⊆V
(G) and E (H) ⊆ E (G).

1.f)How many edges are there in a graph with 10 vertices
each of degree 6 ? [June 2014,3 marks]
According to Handshaking theorem,
q: number of edges
p: number of vertices
di: degree of vertex I
In the above question : p=10, d(i)=6
2q=10*6=60
q=30

3.b)Define Isomorphism of two graphs. Find whether the
given graphs are isomorphic or not.[June 2014,5 marks]

Number of vertices in first graph = 8
Number of vertices in second graph = 8
Number of edges in first graph = 10
Number of edges in second graph = 10
Degree sequence of both the graphs is : <3,3,3,3,2,2,2,2>
These conditions satisfies but still the graphs are non-isomorphic.This is because the two
graphs are not structurally identical.
a b
c d
e f
g
h
A B
C D
E F
G
H
G
H

In graph G, A is a vertex of degree 2, which must corresponds to either B, D, F or G in H.
Each of these four vertices in H is adjacent to another vertex of degree two in H, which is not
true for a in G.
Therefore, these are not isomorphic.

5.b)State and prove Handshaking Theorem.[June 2014, 5
marks]

1.d)State and prove Handshaking Theorem.[December
2014,December 2010, 4 marks]

1.a)Define : 4 (i) Graph (ii) Simple Graph (iii) null
graph (iv) connected Graph [December 2013,4 marks]
i) Graph :It is a set of the form {(x,f(x)): x is a domain of function f}.
ii) Simple graph :
Undirected graph that has no loops or multiple edges is called a simple graph.When an edge
joins a vertex to itself is called a loop.Two or more edges that joins the same vertices are
parallel or multiple edges.

iii) Null graph : A graph with isolated vertices and no edges is called a null graph. It is also known
as an empty graph.
iv) Connected graph : A graph is connected when there is a path between every pair of vertices.
In a connected graph, there are no unreachable vertices.

1.d)Define δ (G) and Δ (G) for a graph G.[December 2013,2
marks]
δ(G) is called the minimum vertex degree of G. It is min{dG(x) :x∈V(G)} . It is a non-negative
integer.
δ(G)=1
Δ (G) is called the maximum vertex degree of G. It is max{dG(x) :x∈V(G)}. It is a non-negative
integer.
Δ(G)=3

4.b)Are the following graphs isomorphic ? If Yes or No
justify.[December 2013,June 2010,4 marks]
Number of vertices in G = 6
Number of vertices in H = 6
Number of edges in G = 11
H
G

Number of edges in H = 10
The two graphs have different number of edges.Therefore, the two graphs are not isomorphic.

1.d)Find the degree of each vertex in the given
graph.[June 2012,4 marks]
Degree of each vertex in the above graph is :
d(v1)=2 d(v6)=
d(v2)=4 d(v7)=2
d(v3)=4
d(v4)=2
d(v5)=3

1.e)What is the complement of the given graph.[June 2012,4
marks]
Complement of the above graph :

2.a)Determine whether the graphs are isomorphic. [June
2012,5 marks]
G H
V(G)=5 V(H)=5
E(G)=8 E(H)=7
Number of edges is not the same in G and H.
Therefore, the graphs are not isomorphic.

1.b)Construct a 5-regular graph on 10 vertices.[December
2012,June 2010,3 marks]

1.b)A graph G is said to be self complementary if it is
isomorphic to its complement G . Show that for a self
complementary (p, q)-graph G, either p or (p – 1) is
divisible by 4.[June 2011,4 marks]
SupposeG is a (p, q)-graph.
Then E (G) ∪ E( ) = {the set of all pairs of vertices inV (G)}.
Thus, q+ =(p(p-1))/2
If the graph G is self complementary, then q = .Thus, p (p –1) = 2q + 2q = 4q, that is 4 divides
p (p –1). Since only one of p or (p –1) is even, this means either p or (p – 1) is divisible by 4.

1.c)Define minimum vertex degree of G (δ (G))and maximum
vertex degree of G (Δ (G)).[June 2011,3 marks]
δ(G) is called the minimum vertex degree of G. It is min{dG(x) :x∈V(G)} . It is a non-negative
integer.
δ(G)=1
Δ (G) is called the maximum vertex degree of G. It is max{dG(x) :x∈V(G)}. It is a non-negative
integer.
Δ(G)=3

5.a)Can a simple graph exist with 15 vertices, with each
of degree five ? Justify your answer.[June 2011,3 marks]
A corollary in graph theory states that “Any graph can only have an even number of odd
vertices”.This is because of the handshaking problem.
According to the question,
Number of vertices, p=15
Degree of each vertices, D(Vi)=5
This graph has 15 odd vertices which is odd, so the above graph cannot exist.

5.b)Are the following graphs are isomorphic ? 4 If Yes or
No Justify.[June 2011,4 marks]
G
H

Degree Sequence of H : <2,2,2,2,2>
f(a)=2 f(b)=4 f(c)=1 f(d)=3 f(e)=5
This shows that the two graphs are isomorphic.

4.a)Define the concept of a complete graph. Draw complete
graph each for the case when number of vertices is given
by : n=3, n=4.[June 2010,3 marks]
Complete graph : Graph in which any two vertices are adjacent, i.e. each vertex is joined to
every other vertex by a vertex. A complete graph on n vertices is represented by Kn.
n=3
n=4

1.c)Define r-regular graph. Give an example of 3-regular
graph.[December 2010,3 marks]
Kn is a regular graph with degree of regularity (n-1) when n > 3.

1.b)Show that the sum of the degrees of all vertices of a
graph is twice the number of edges in the graph.[June
2009,3 marks]
Sum of the degrees of al vertices of a graph is twice the number of edges in the graph.This is
called handshaking problem.

1.c)Define isomorphism of graphs. Determine whether the
graphs are isomorphic.

Degree sequence of G : <3,2,2,1,1,1>
Degree sequence of H : <3,2,2,1,1,1>
The two graphs are not isomorphic.This is because in graph G, vertex p with degree 3 is
adjacent to two vetices of degree 1 (u,v) and a vertex with degree 2 (q).This is not the case in
graph H(vertex with degree 3 is adjacent to two vertices with degree 2 and a vertex with degree
1).

1.f) What is the complement of the given graph?[June
2009,3 marks]

3.b)How many vertices will the following graphs have if
they contain :[June 2009,4 marks]
i) 16 edges and all vertices of degree 2.
Sum of all degrees of vertices=2* number of edges
Let number of vertices be n.
2 * n= 2*16
n=16
ii) 21 edges, 3 vertices of degree 4 and the other vertices of degree 3
Let n be the number of vertices.
(3*4) +(n*3)=2*21
12+3n=42

1.b)The number of vertices of odd degree in a graph is
always even.[December 2009,3 marks]
Any graph can only have an even number of odd vertices.
Consider a (p,q) graph with {x1,x2,…..xt} is a set of odd vertices and {xt+1,…..xp} is a set of
even vertices.
Let dG(xi)=2ci+1 1≤i≤t and dG(xi)=2ri t+1≤i≤p

1.c)What is the complement of the given graph?[December
2009,2 marks]

4.b)What is the largest number of vertices in a graph with
35 edges if all vertices are of degree at least 3
?[December 2009,5 marks]
Maximum degree of a graph >= Sum of degree of individual vertices
2E >= deg(V1) + deg(V2) + ... + deg(Vn)
2 * 35 >= 3 + 3 + ... + 3 ...(I),
70 >= 3n
23.33 >= n or 23 >= n

1.a)Consider the graph below :[June 2008,2 mark]
i) Find δ (G) and , Δ (G)
Degree of v1 = 3 Degree of v2 = 3 Degree of v3 = 3 Degree of v4 = 4 Degree of v5 = 3
Degree of v6 = 4 Degree of v7 = 4 Degree of v8 = 4
From the above diagram, δ (G) = 3 Δ (G) = 4

ii) Draw the subgraph induced by the set {v1,v6, v4, v7,v2}

3.c)Draw a 4-regular graph on 6 vertices.[June 2008,2
marks]

1.a)Show that the graphs G and G’ are isomorphic.[December
2008,4 marks]
Number of vertices in G=7 Number of vertices in G’= 7
Number of edges= 13 Number of edges= 14
Therefore both the graphs are not isomorphic.

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