Discrete Mathematics Please show work if required proof is n

Discrete Mathematics:

Please show work if required proof is needed for the answer. There are 3 parts to this question:

True or False (circle your answer):

a)

b)

c)

True False: There is a graph with degree sequence 0, 1,2, 3

Solution

Solution :- a) we have given degree sequence 0,1,2,3.

But this sequence is not graphic. If there were a graph G with degree sequence (0, 1, 2, 3), then G would have 4 vertices, one of which (the one with degree 0) is not adjacent to any other vertex, and one of which (the one with degree 3) is adjacent to every other vertex. This is a contradiction.

So False.

b) True, because there are different 26 aplhabets, we have to choose 2 of them.and we have 9 different digits , we have to choose 2 of 9.

So we get the answer true.

C) We have a_n = a_n-1 + 2n -1

The associated homogeneous equation is a_n = a_n-1

the charactaristc equation is r - 1 = 0 so r = 1  and so its solution is a_n^(h) = 1ⁿ , where is a constant.

F(n) = 2n - 1 ,so there is a particular P(n) solution is of the form p₁ n + p₀

Now substitute this in the recurrence equation

p₁n + p_{0 =} p₁(n-1) + p₀ +2(n -1)-1

p₁n + p_{0 =} p₁n - p₁ + p₀ + 2n - 2 - 1

0 = - p₁ + 2n - 3

So p₁ = 2n - 3

p₀=0

So the solution of the given recurrence equation is a_n = 1ⁿ -2n - 3

So the given statment is false.