> #Weiji Zheng,  Assignment 18, 11/22/2020
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> #OK TO POST HOMEWORK
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> 
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> #Q1
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> #Generalize procedure NuConG(N) of M18.txt to write a procedure NuKcomponentsGraphs(N,K)
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> #that inputs a positive integer N and another positive integer K and outputs the first N terms of the sequence enumerating the number of labeled graphs on n vertices with Exactly K componets. Make sure that NuKComponentsGraphs(N,1) equals NuConG(N) for N ¡Ü 6
> #Hint: Of course, you first need to write a procedure AllKcomponentsGraphs(n,K) by tweaking procedure AllConGraphs(n) and change the line nops(CCs(s))=1 to nops(CCs(s))=k
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> #Are the following sequences in the OEIS? If yes, what are the A-numbers NuConGgen(5,2), NuConGgen(5,3)
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> 
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> #AllKcomponentsGraphs(n,K)
> AllKConGraphs:=proc(n,K) local s,S:
> S:={};
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> for s in AllGraphs(n) do
>     if nops(CCs(s))=K then
>        S:= S union {s}
>     fi:
> od:
> S:
> end:
> 
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> #NuKcomponentsGraphs(N,K)
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> NuKcomponentsGraphs:=proc(n,K) local nn:[seq(nops(AllKConGraphs(nn,K)),nn=1..n)]:end:
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> NuConG(4):
[1, 1, 4, 38]
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> NuKcomponentsGraphs(4,1)
[1, 1, 4, 38]
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> 
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> NuKcomponentsGraphs(5,2)
[0, 1, 3, 19, 230]
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> #A323875
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> 
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> NuKcomponentsGraphs(5,3)
[0, 0, 1, 6, 55]
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> #A323876
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> 
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> #Q2
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> #Using procedures RandGr(n,K) and CCs(G), in M18.txt write a procedure AveNuCC(n,K,M) that generates M random graphs on n vertices with K edges, and for each of them finds the number of components and take the average over these M random graphs. Run it several times with n=30 and M=1000 to check that you get close answers. If not what M would give such agreement?
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> 
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> AveNuCC:=proc(n,K,M) local a,total,res:
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> total := 0:
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> for a from 1 to M do
>     total := total + nops(CCs(RandGr(n, K))):
> od:
> res := total/M:
> res:
> end:
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> 
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> #Q3
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> #Using the above,say, write a procedure EstimateCutOff(n,M)that inputs a positive integer n and outputs an estimate for the "threshold" for K, when most graphs with K edges start to be connected (say when the average number of connected componets is less than 1.05)
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