Question 1.
How many labeled connected graphs with 50 vertices and 70 edges are?

Answer 1.
We use the wtEdgConClever proc in maple to find this
557570125262936299552095051452832858951958306952392379552719516748979534894703561052813706583449695070191616000000
Number of label connected graphs with 50 vertices and 70 edges

Question 2. 
How many labeled graphs with i connected components with 50 vertices and 70 edges are for i=2,3,4,5? Explain how you do it, by using the taylor command in Maple.

Answer 2. 
I wrote a new procedure to do this
WtEdConGclever2:=proc(N,a, k) local f,i,x:
option remember:
f:=log(add((1+a)^binomial(i,2)*x^i/i!,i=0..N+1))^k/k!:
f:=taylor(f,x=0,N+1):
[seq(expand(coeff(f,x,i)*i!),i=1..N)]:
end:

With 2 components
2267762282072205271778791270680596849921516338574571823883456519281730035101965414722739023408444177334403072000000

With 3 components
4084614615513849564059280183743190588234377301848961646892154560205839868330317696952259164363869748482983526400000

With 4 components
4336520950646656327224993268880264842114984745013581637572959168612209444462829083123209142308423023328834795520000


Question 3
Compare Estimate cutoff and findCutoff

Answer 3
EstimateCutOff(30, 1000)
Pretty close
                               81
FindCutOff(30, 1.05)
                               84

Question 4
Try to do the optional challenge from the homework of Lecture 18 exactly, using evalf(FindCutOff(n,1.05)/n), for n= 20,30,40,50,60. Do you see a trend?

Answer 4
seq(evalf(FindCutOff(n, 1.05)/n), n=20..60,10)
     2.450000000, 2.800000000, 3., 3.160000000, 3.300000000

It's an increasing trend

Question 5
Generalize procedure
EstProbConn(n,m,N):

and write a procedure

EstProbKcomps(n,m,k,N):

that estimates the probability that a random graph with n vertices and m edges has exactly k components, by sampling N such random graphs.

Note that EstProbKcomps(n,m,1,N) is exactly the same as EstProbConn(n,m,N) (except, since we are doing simulations, you get different answers each time)
Answer 5
EstProbKConn:=proc(n,m,N, k) local i,co, G:

co:=0:

for i from 1 to N do

G:=RandGr(n,m):

if nops(CCs(G))=k then
 co:=co+1:
fi:

od:

evalf(co/N):
end:


Question 6
ProbConn(n,m):

To write a procedure

ProbKcomponents(n,m,k)

that uses the method of exponential generating functions to find the EXACT probability that a random graph with n vertices and m edges has k components. Note that ProbKcomponents(n,m,1) is EXACTLY the same as ProbConn(m,m,1)

How does ProbKcomponents(40,50,2) differ from EstProbKcomps(40,50,2,300)?
Answer 6
ProbKComponents:=proc(n,m, k)  local a:
evalf(coeff(WtEdConGclever2(n,a, k)[n],a,m)/binomial(n*(n-1)/2,m)):
end:


ProbConn(10, 20)
                          0.9768915520
EstProbKConn(40, 50, 300, 2)
                         0.09666666667