Graph theory

Question

Answer 1

it should be 2^(n-2)

Answer 2

In K_n, every vertex is connected by other vertices through an edge. This question can be easily solved by checking for n=2,3,4,5....vertices. for n=2, strictly increasing path is 1, for n=3, it is 2, for n=4, it is 4 and so on. so the total no. of strictly increasing path between V₁ and V_n is 2^n-2.

Answer 3

Vertices are labeled as 1,2,3,......n. Now for each path 1st vertex will be 1 and the last vertex will be n.So the remaining n-2( labeled 2,3,....n-1) vertcies could be internal vertices of the path.

Now  the number of choices of the internal vertices these could be the number of subsets of the set(2,3,....n-1).

So the answer will be 2^n-2. So B