#C2.txt: Jan. 27, 2025
Help2:=proc(): print(`LC(p), RG(n,p), Cliques(G,k) `):end:
with(combinat):
#LC(p): inputs a rational number between 0 and 1 and outputs true with prob. p
LC:=proc(p) local a,b,ra:
if not (type(p,fraction) and p>=0 and p<=1) then
 RETURN(FAIL):
fi:
a:=numer(p):
b:=denom(p):
ra:=rand(1..b)():
if ra<=a then
 true:
else
 false:
fi:
end:

RG:=proc(n,p) local E,i,j:
E:={}:
for i from 1 to n do
 for j from i+1 to n do
   if LC(p) then
    E:=E union {{i,j}}:
   fi:
 od:
od:
[n,E]:
end:

#Cliques(G,k): inputs a graph G and a pos. integer k outputs the set of
#k-cliques
Cliques:=proc(G,k) local n, E,S,i,c,C:
n:=G[1]:
E:=G[2]:
S:={}:
C:=choose({seq(i,i=1..n)},k):
for c in C do
 if choose(c,2) minus E={} then
  S:=S union {c}:
 fi:
od:
S:
end:

###From C1
#C1.txt: Jan. 23, 2025 Exp Math (Dr. Z.)
Help1:=proc(): print(`Graphs(n), Tri(G) , TotTri(G) `):   end:



#An undirected  graph is a set of vertices V and a set of edges
#[V,E]   and edge e={i,j} where i and j belong to V
#Our vertices are labeled {1,2,...,n}
#Our data structure is [n,E] where E is the set of edges
[3,{{1,2},{1,3},{2,3}}];

#If there are n vertices how many (undirected) graphs there

#Graphs(n): inputs a non-neg. integer and outputs the set of ALL
#graphs on {1,...,n} 
Graphs:=proc(n) local i,j,S,E,s:
E:={seq(seq({i,j},j=i+1..n),   i=1..n)};
S:=powerset(E):
{seq([n,s],s in S)}:
end:

#Tri(G): inputs a graph [n,E] and outputs the set of all triples {i,j,k}
#such {{i,j},{i,k},{j,k}} is a subset of E
Tri:=proc(G) local n,S,E,i,j,k:
n:=G[1]:
E:=G[2]:
#S is the set of love triangles
S:={}:
for i from  1 to n do
 for j from i+1 to n do
   for k from j+1 to n do
    #if member({i,j},E) and member({i,k},E), and member({j,k},E) then
     if {{i,j},{i,k},{j,k}} minus E={} then
       S:=S union {{i,j,k}}:
    fi:
   od:
 od:
od:

S:
end:

#Comp(G): the complement of G=[n,E]
Comp:=proc(G) local n,i,j,E:
n:=G[1]:
E:=G[2]:
[n,{seq(seq({i,j},j=i+1..n),   i=1..n)} minus E]:
end:
#Tot(G): the total number of love triangles and hate triangles
TotTri:=proc(G)
nops(Tri(G))+nops(Tri(Comp(G))):
end: