Answer to Question #306170 in Algebra for samy

Question

Find all trees T where 75% of the vertices of T have degree 1 and the remaining 25% of the vertices have another degree.(a fixed degree)

Answer 1

Let T be a tree of order n where 75%of the vertices have degree 1 and the remaining 25% have a fixed degree X.Then (3n/4).1+(n/4).X=(2n-1).Then n(5-x)=8 implying that X ≥4 so X=2,3,4 since is an integer ,X ≠2.Thusx=3 X=4.if X=3,then n=4 and T=k↓¹³:while if X=4 then n=8 and T is the double star in (a)

(a)

Π=summation of I Π¡ Since

and

2(n-1) =2m=summation of ¡ ¡Π¡