#ATTENDANCE QUIZ FOR LECTURE 21 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p21
#with an attachment called
#p20FirstLast.txt
#(e.g. p21DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 20, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 4

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER


QUESTION #1:

How many connected Graphs are there with 30 vertices and 50 edges?

ANSWER:

L:=WtEdConGclever(35,a); gives a sequence of weight-enumerators where the ith element the weight-enumerator for i vertices.
So, to find the number of connected graphs with 30 vertices and 50 edges we do:
coeff(L[30],a,50):
616531513027177775473601317524937092680978914590598004251998471840 is the number of connected graphs with 30 vertices and 50 edges.

QUESTION #2:

Use NuKcomp(N,k) to find the number of labeled graphs with 5 components and 50 vertices

ANSWER:

Running NuKcomp(50, 5)[50] we get:
84788987690871322213720759813193437723400498985997782717718839577530578491639411074799027505529978132256881827900900288575885348212857384767830040557053247256490676318714477784265805319979127545883387577706020661320917668889000205517435779132191852077004426477271855375638163003674948760253727467070410930283337482240
is the number of labeled graphs with 5 components and 50 vertices

QUESTION #3:

How many labeled connected graphs are there with 30 vertices and 50 edges?

ANSWER:

This is the same question as #1, where we found the answer to be:
616531513027177775473601317524937092680978914590598004251998471840 graphs

QUESTION #4:

For i = 2,3,4,5,... Find the OEIS A-number of NuKcomp(20,i). What is the smallest i for which the sequence is not in the OEIS?

ANSWER:

Running NuKcomp(20,i)...

For i=1:
[1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896, 1328578958335783201008338986845427712, 87089689052447182841791388989051400978432, 11416413520434522308788674285713247919244640256, 2992938411601818037370034280152893935458466172698624, 1569215570739406346256547210377768575765884983264804405248]

Its A-number is A001187.

For i=2:
[0, 1, 3, 19, 230, 5098, 207536, 15891372, 2343580752, 675458276144, 383306076989440, 430041136692146912, 956431386434331323776, 4224539434553753578497024, 37106501188130085159785113344, 648740172906485727983524271405824, 22591360806791558877526051411343415296, 1567817808096346724727108606144936617617408, 216926754380646324945248257231539109504762560512, 59860937747979290094875740320161640032623120090828800]

Its A-number is A323875.

For i=3:
[0, 0, 1, 6, 55, 825, 20818, 925036, 76321756, 12143833740, 3786364993664, 2323363153263768, 2810644049356050752, 6714880790313869814368, 31734660624638397560681792, 297106568651256947892439231872, 5516820501457062391874183605225216, 203371936690880564729559424288326233856, 14896201998273652941883043518617399703696384, 2169416538066466491819023076937523996727138210304]

Its A-number is A323876.

For i=4:
[0, 0, 0, 1, 10, 125, 2275, 64673, 3102204, 272277040, 46202044900, 15442093276764, 10171924771814520, 13188852179018387144, 33674263441006260931040, 169522275849148918884400912, 1685048703908907788901122512512, 33116110237646373502366665503208064, 1288337109916947580133035603563656989952, 99320901948403913391024993536094346775110656]

Its A-number is A323877

The smallest i such that NuKcomp(20,i) is not found in the OEIS is i=5.