#OK to post homework
#Joseph Koutsoutis, 01-26-2025, Assignment 1

read `C1.txt`:

Help := proc(): 
 print("AM(G), Image(pi,G), ULgraphs(n)"):
end:

#2 

# Exercise 1.1a

# By the axioms for inner products, we find:
# {<A| + <B|} |C> = [{<A| + <B|} |C>]**
#                 = [<C| {|A> + |B>}]*
#                 = [<C|A> + <C|B>]*
#                 = <C|A>* + <C|B>*
#                 = <A|C> + <B|C>

# Exercise 1.1b

# By the axioms for inner products, we have that:
# <A|A>* = <A|A>
# It follows that <A|A> is a real number.

# Exercise 1.2

# We first find:
# <C| {|A> + |B>} = (c1*)(a1 + b1) + ... + (cn*)(an + bn)
#                 = (c1*)(a1) + ... + (cn*)(an) + (c1*)(b1) + ... + (cn*)(bn)
#                 = <C|A> + <C|B>
# so we have linearity.

# We next check:
# <B|A> = (b1*)(a1) + ... + (bn*)(an)
#       = [(b1*)(a1) + ... + (bn*)(an)]**
#       = [(b1)(a1*) + ... + (bn)(an*)]*
#       = [(a1*)(b1) + ... + (an*)(bn)]*
#       = <A|B>*
# so interchanging bras and kets corresponds to complex conjugation.


#3 

AM := proc(G) local n,E,i,j,M,e:
  n,E := op(G):
  M := [seq([seq(0, j=1..n)],  i=1..n)]:
  for e in E do:
    i,j := op(e):
    M[i][j] := 1:
    M[j][i] := 1:
  od:
  M:
end:


#4

Image := proc(pi, G) local i,j,n,E1,E2,e:
  n,E1 := op(G):
  E2 := {}:
  for e in E1 do:
    E2 := E2 union { {op(pi[[op(e)]])} }:
  od:
  [n,E2]:
end:


#5

ULgraphs := proc(n) local all_graphs,G,pi,pis,representatives,seen:
  all_graphs := Graphs(n):
  pis := permute([seq(1..n)]):
  representatives := {}:
  seen := {}:
  for G in all_graphs do:
    if not G in seen then:
      representatives := representatives union {G}:
      for pi in pis do:
        seen := seen union {Image(pi, G)}:
      od:
    fi:
  od:
  
  representatives:
end:

# I found that [seq(nops(ULgraphs(i)),i=1..6)] outputs [1, 2, 4, 11, 34, 156].
# This matches A000088 (https://oeis.org/A000088), which is the number of simple graphs on n unlabeled nodes.