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



with(combinat):

#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: