The Gateway to Computer Science Excellence
+31 votes
4.1k views

How many undirected graphs (not necessarily connected) can be constructed out of a given set $V=\{v_1, v_2, \dots v_n\}$ of $n$ vertices?

  1. $\frac{n(n-1)} {2}$
  2. $2^n$
  3. $n!$
  4. $2^\frac{n(n-1)} {2} $
in Graph Theory by Veteran (52.2k points)
edited by | 4.1k views
0
The answer corresponds to graph without multiple edges and self loops i.e. simple graphs.

@Bikram Am I correct.
+8

Shivam Chauhan

yes, generally we consider simple graphs, if nothing is mention .

+2

@Bikram Sir
I understood the solution. But, why is

nC0 + nC1 + nC2   . . . . nCn = 2n a wrong answer ?

Since, single vertex graph is also a connected graph and there will be n such graph. Now if we take 2 vertex out of n then we form nC2 graphs and so on . .

What mistake am I committing here ?

+2
Every selection of vertex will also include multiple graphs.

Ex you have chosen 3 nodes in nC3 ways, now u can again form many graphs from it. That's why this answer is wrong
0
^^

Okay, I got it. Thank you.

3 Answers

+47 votes
Best answer
With $n$ vertices we have max possible $^{n}C_{2}$ edges in a simple graph. and each subset of these edges will form a graph, so total number of undirected  graph possible = $2^{\frac{n(n-1)}{2}}$

Correct Answer: $D$
by Boss (13.5k points)
edited by
+30
Another way:

In a complete graph there are [n*(n-1)]/2 edges.

Any of the edge can be selected/neglected, so  2^([n*(n-1)]/2)
+1
Can you please tell the case when I have only few vertices like only v1 ,v2,v3 ,Now this is not counted in these cases as we are just counting the total no of subsets of edges , and the case where we do not chose any edge there we shall have all the vertices isolated from each other , but how to  consider the case where we may have fewer vertices and they still remain isolated .
0
Sorry did nt get ur question ?? can u elaborate it ??
0

I have doubt in this answer , in this question it is not mentioned that graph is labeled. 

hence for n=3 above answer gives number of graphs as 8. But in reality they are 4.

Please refer answer of this question , https://gateoverflow.in/2443/gate1994_1-6-isro2008-29 

Your answer could be correct if graph is labeled 

+2
@mehul in question vertices are labeled here. V={v1,v2---vn} that's why we are considering labeled graph.
0
Great explanation..thnx
+1
What if they had asked “how many CONNECTED graphs can be constructed” ?
+2 votes

No. of vertices in the given questions $= n$

In an undirected simple graph, the maximum no. of edges that are possible $= n(n-1)/2$

Now, each edge can be either present or absent in a graph. So, there are 2 possibilities for each vertices.

Therefore, total no. of possible graphs $= 2*2*2*...  n(n-1)/2$ times

=$2^{n(n-1)}$ 

So, correct answer is option no. D.

by Junior (809 points)
0 votes

I have doubt in this answer , in this question it is not mentioned that graph is labeled. 

hence for n=3 above answer gives number of graphs as 8. But in reality they are 4.

Please refer answer of this question , https://gateoverflow.in/2443/gate1994_1-6-isro2008-29 

Your answer could be correct if graph is labeled 

by Active (4.9k points)
0
can u help me how u find the value of n(n-1)/2.??plese explain i cant understand
0
then what is v1,v2,v3,....vn.??

are not they labels...read carefully
0

@mehul vaidya In gate 1994 question asked upto 3 nodes i.e you consider 0 node , 1 node , 2 nodes and 3 nodes.

But Question based on unlabeled graphs because they asked upto 3 nodes

     so for and 3 nodes we construct 4 different unlabeled graphs

As 3 vertex with 3 edges = 1 graph

  3 vertex with 2 edges = 1 graph

3 vertex with 1 edge = 1 graph

3 vertex with 0 edge = 1 graph

similarly 2 nodes we construct 2 different unlabeled graphs

2 vertex with 2 edges = 0 graph

2 vertex with 1 edges = 1 graph

2 vertex with 0 edges = 1 graph

and  1 nodes we construct only 1 unlabeled graphs

 1 vertex with 1 edges = 0 graph

1 vertex with 0 edges = 1 graph

these are the total 7 unlabeled graphs construct

and out of these 7 graphs only 4 graphs are connected....

which are  

3 vertex with 3 edges = 1 graph

  3 vertex with 2 edges = 1 graph

2 vertex with 1 edges = 1 graph

1 vertex with 0 edges = 1 graph  ( single node also called as connected graph in 1 vertex graph ).

if you considering Labeled graph then formula is 2^n(n−1)/2

for 3 node = 2^3(3-1)/2 = 8 labeled graphs construct

for 2 nodes = 2 labeled graphs construct

for 1 node = 1  labeled graphs construct

so total 11 graphs generates , if Labeled graph taken.

I hope now you cleared doubts.

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,647 questions
56,461 answers
195,358 comments
100,240 users