Question 9015 – GATE 2010

i_d= no. of vertices of degree ‘d’ in ‘G’
Eg:

No. of vertices with degree ‘2’ = 3
ξ(G’) = 3 × 2 = ‘6’ i.e., sum of degrees

By Handshaking Theorem,
The sum of degrees would be equal to twice the no. of edges
|V| = 2|E|

It is given that ξ(G) = ξ(S) then

Sum of degrees of vertices in G is equal to sum of degrees of vertices in S
i.e., 2*(no. of edges in G) = 2*no. of edges in S
no. of edges in G = no. of edges in S
Eg:

ξ(G) = (2 × 2) + (2 × 3) = 4 + 6 = 10

ξ(S) = 2 × 5 = 10

You can observe that, though no. of vertices are different, but still no. of edges are same.