A planar drawing of a connected graph G𝐺 has four faces, whose degrees are 3, 4, 5 and 8 respectively.How many edges does the graph have? Answer 1 Question 2How many vertices does the graph have?

Question 1:

The number of edges in a planar graph can be calculated using the formula for the sum of the degrees of the faces, which is 2E, where E is the number of edges.

Given that the degrees of the faces are 3, 4, 5, and 8, the sum of the degrees of the faces is 3 + 4 + 5 + 8 = 20.

Therefore, 2E = 20, and solving for E gives E = 20 / 2 = 10.

So, the graph has 10 edges.

Question 2:

The number of vertices in a graph can be calculated using Euler's formula for planar graphs, which is V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.

We know from the previous question that E = 10, and we know from the question that F = 4.

Substituting these values into Euler's formula gives V - 10 + 4 = 2, and solving for V gives V = 10 - 4 + 2 = 8.

So, the graph has 8 vertices.