Question

Q 10. A simple graph have 14 edges, 3 vertices of degree 4, 2 vertices of degree 6 and all others of degree 2.
How many vertices does the graph have?
Ops: A. 09
B. O 10
C.
08
D. 07
Ans?

Answer 1

Answer:

The first theorem of Graph Theory says that the sum of all vertex degrees equals twice the number of edges. So if there are n vertices in the graph, then (3⋅4)+2(n−3)=2⋅9=18 . This gives n=6 . ■

Answer 2

The number of vertices in the graph = 7

Given :

A simple graph have 14 edges, 3 vertices of degree 4, 2 vertices of degree 6 and all others of degree 2.

To find :

The number of vertices in the graph

A. 9

B. 10

C. 8

D. 7

Concept :

For a simple graph sum of all degree = 2 × Number of edges

Solution :

Step 1 of 2 :

For the equation to find number of vertices in the graph

Here it is given that the simple graph have 14 edges

∴ The sum of all degree

= 2 × Number of edges

= 2 × 14

= 28

Now it is given that 3 vertices of degree 4, 2 vertices of degree 6 and all others of degree 2.

Number of vertices of degree 4 = 3

Number of vertices of degree 6 = 2

Let there are n vertices of degree 2

∴ The sum of all degree

= (3 × 4) + (2 × 6) + (n × 2)

= 12 + 12 + 2n

= 2n + 24

By the given condition

$\displaystyle \sf{ 2n + 24 = 28}$

Step 2 of 2 :

Find number of vertices in the graph

$\displaystyle \sf{ }2n + 24 = 28$

$\displaystyle \sf{ \implies 2n = 28 - 24}$

$\displaystyle \sf{ \implies 2n =4}$

$\displaystyle \sf{ \implies n = \frac{4}{2} }$

$\displaystyle \sf{ \implies n = 2}$

Number of vertices of degree 4 = 3

Number of vertices of degree 6 = 2

Number of vertices of degree 2 = 2

∴ The total number of vertices in the graph

= 3 + 2 + 2

= 7

Hence the correct option is D. 7

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