#hw1SalmanManzoor.txt



with (combinat):



#AM(G): takes as input a graph [n,E] and outputs the adjacency matrix,

#represented as a list of length n of lists of length n, 

#such that M[i][j]=1 if {i,j} belongs to E and 0 otherwise.



AM:=proc(G) local n,E,i,j,A:

n:=G[1]:

E:=G[2]:

A:=[[0$n]$n]:

for i from 1 to n do

	for j from 1 to n do

		if member({i,j}, E) then

			A[i][j]:=1:

		fi:

	od:

od:

A;

end:



#Image(pi,G): takes as input a permutation pi of {1,...,n} and

#a graph G=[n,E] and outputs the image under pi



Image:=proc(pi,G) local n,k,E,i:

n:=G[1]:

E:=G[2]:

k:=nops(E):

for i from 1 to k do

	E[i]:={pi[E[i][1]], pi[E[i][2]]}:

od:

[n,E];

end:



#ULgraphs(n): outputs a set of graphs with ONE member of each equivalence class

#of labelled graphs under the equivalence relation of graph isomorphism for 

#the given number of vertices



ULgraphs:=proc(n) local i,j,k,S,E,s,H,P,F,D,p:

E:={seq(seq({i,j},j=i+1..n),   i=1..n)};

S:=powerset(E):

H:={seq([n,s],s in S)}:

P:=permute(n):

F:={}:

p:=length(P):

do

	D:=H[1]:

	F:=F union {D}:

	for k from 1 to p do

		H:=H minus {Image(P[k],D)}:

	od:

until isEmpty(H):

	

end:



#[seq(nops(ULgraphs(i)),i=1..6)] yields [1,1,2,4,11,34] which aligns with

#sequence A000088 



#{<A| + <B|} |C> = <C|{|A> + |B>}* = ( <C|A> + <C|B> )* = <A|C> + <B|C>



#From <A|A> = <A|A>* , we see <A|A> must be a real number to be equal to its own conjugate.



#First axiom follows from distributivity of complex multiplication. a(b+c) = ab+ac for complex a,b,c. So the 2n terms of <C|A> + <C|B> can be combined to form the n terms of <C|{ |A> + |B>} .



#Second axiom follows from linearity of the complex conjugate, (ab + cd)* = (a*b* + c*d*)