Pease prove the following e 2044Solutiona we are given that

Pease prove the following:

(e) 20->(4,4).

(a) we are given that a,b>=2 and n>=2 as well

since n is the number of vertices of graph G

and alpha(G)>=a

and omega(G)>=b

=> n>=a and n>=b

and when n>=2 its possible to have a=2 and b =2

hence it n>=2 , n --> (2,2) proved

(b)

now n>=2 and a>=2 so

n=a could be possible

this\'ll happen when the number of vertices = independence number

and we know that b>=2 it true

therefore for n>=2 , n --> (n,2)

hence proved

(c)

m>=n

we are given that n --> (a,b)

and when the number of vertics of graph G equals m

then is fine to say that m --> (a,b)

when m>=2 and n --> (a,b)

hecne roved

(d)

given n --> (a-1,b) and m --> (a,b-1)

now (n+m) --> (2a-1 , 2b-1)

we are given that a,b>=2

=> (n+m) --> (4-1, 4-1)

(n+m) --> (3,3)

thi shows that a,b>=3

therefore its true to say that (n+m) --> (a,b)

(e)

we have already proved that n --> (a,b)

and we are given that n,a,b >=2

when the graph G has n=20 vertices then the independence number could be = a=4 and the clique number = b =4

therefore 20 --> (4,4)

hence proved