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Home / Discrete mathematics / A tree has two vertices of degree 2. One vertex of degree 3 and 3 vertices of degree 4.How many vertices of degree 1 does it have?

A tree has two vertices of degree 2. One vertex of degree 3 and 3 vertices of degree 4.How many vertices of degree 1 does it have?

Exercise:A tree has 2n vertices of degree 1. 3n vertices of degree 2 and n vertices of degree 3. Determine the number of vertices and edges in the tree.

Solution:

2n vertices has total degree = 2n× 1 = 2n

3n       ″      ″    ″      ″   = 3n×2 =6n

n       ″      ″    ″      ″    = n×3 =3n

6n       ″      ″    ″      ″               =11n       [ By adding]

Since the sum of the degree of the vertices of a tree with n vertices is

2(n-1) = 2n-2

∴ 11n = 2.6n-2

⟹11n=12n-2

⟹12n-11n=2

⟹ n=2

Therefore the total number of vertices is v= 6n = 6×2=12 (Answer)

And the total number of edges is e = v-1 = 12-1 = 11 (Answer)

 

Exercise:A tree has two vertices of degree 2. One vertex of degree 3 and 3 vertices of degree 4. How many vertices of degree 1 does it have?

Solution:

2 vertices has total degree = 2× 2 = 4

1       ″      ″    ″      ″   = 1×3 =3

3       ″      ″    ″      ″   = 3×4 =12

n       ″      ″    ″      ″    = n×1 =n

6+n       ″      ″    ″      ″              =19+n       [ By adding]

Since the sum of the degree of the vertices of a tree with n vertices is

2(n-1) = 2n-2

∴19+n= 2. (6+n) -2

⟹19+n=12+2n-2

⟹12+2n-2=19+n

⟹ 2n-n=19+2-12

⟹n=9

Hence the number of vertex of degree 1 is 9. (Answer)

 

 

 

 

 

 

 

 

 

 

 

 

 

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