UGC NET CSE | December 2006 | Part 2 | Question: 3

Question

The number of edges in a complete graph with $N$ vertices is equal to :

• A. $N (N−1)$
• B. $2N−1$
• C. $N−1$
• D. $N(N−1)/2$

Comments

Video Explanation: https://youtu.be/EbX7TV0ao6o  (From 05:23 timestamp )

Answer 1

ans is  D

in complete graph there is an edge between every  pair of vertices. so in complete graph with n vertices the degree of each vertex is n-1 . so total degrees of all vertices n(n-1)

according to handshaking theorem

2x No of edges =sum of degree of all vertices (n(n-1) here)

so No of edges =n(n-1)2

Answer 2

In a complete graph of n vertices every vertex is adjacent to n-1 other vertices 
so degree of each node is $n-1$ and there are $n$ vertices ,which makes $n(n-1)$ as the total degree

By handshaking Lemma we have → (For undirected graphs)
Sum of all the degree of all the vertices =$ 2* |E|$ where |E| represents the no of edges in the graph 
$2*|E|=n(n-1)$
$|E|=n(n-1)/2$

So the correct answer will be option D.