Q. Draw the structures of complete undirected graphs on one, two, three, four and five vertices also prove that the number of edges in an n vertex complete graph is n(n-1)/2.
Ans:
The Graphs are drawn below:
i. Graph for One vertex:-
.A
ii. Graph for Two vertices:-
From the above drawn Graphs we can see
i. When n=1,
Then number of edges becomes
=n(n-1)/2
= 1(1-1)/2
=1(0)/2
=0/2
=0
Therefore, number of edges = 0. ii. When n=2,
Then number of edges becomes
=n(n-1)/2
=2(2-1)/2
=2(1)/2
=2/2
=1
Therefore, number of edges = 1. iii. When n=3,
Then number of edges becomes
=n(n-1)/2
=3(3-1)/2
=3(2)/2
=6/2
=3
Therefore, number of edges becomes = 3.
iv. When n=4,
Then number of edges
=n(n-1)/2
=4(4-1)/2
=4(3)/2
=12/2
=6
Therefore, number of edges becomes = 6.
v. When n=5,
Then number of edges becomes
=n(n-1)/2
=5(5-1)/2
=5(4)/2
=20/2
=10
Therefore, number of edges becomes = 10.