# OK to post homework
# James Betti, 23 Feb 2025, Assignment 9

with(combinat):

CheegerC := proc(G) local V,i,h,candidates,S,u,v,T;
    V := {seq(i,i=1..G[1])};
    h := infinity;
    candidates := {};
    for S in seq(op(choose(V,i)),i=1..floor(G[1]/2)) do
        candidates := {};
        for u in S do for v in V minus S do candidates := {op(candidates),{u,v}} od od;
        T := candidates intersect G[2];
        h := min(h,nops(T)/nops(S));
    od;
    h;
end:

CheckBounds := proc(G);
    d,l := op(lam2(G));
    h := CheegerC(G);
    J := [(d-l)/2,sqrt(2*d*(d-l))];
    if h<J[1] or h>J[2] then
        RETURN(FAIL);
    fi;
    [evalf(h),J];
end:

for i from 5 to 10 do
    printf("Gnd(%d,2):",i);
    CheckBounds(Gnd(i,2));
    printf("Gnd(%d,3):",i);
    CheckBounds(Gnd(i,3));
od;

# Gnd(5,2):  [3.,  [2.500000000, 6.324555320]]
# Gnd(5,3):  [3.,  [2.500000000, 6.324555320]]
# Gnd(6,2):  [2.,  [2.000000000, 5.656854249]]
# Gnd(6,3):  [3.,  [3.000000000, 7.745966692]]
# Gnd(7,2):  [2.,  [1.599031132, 5.058112110]]
# Gnd(7,3):  [4.,  [3.500000000, 9.165151390]]
# Gnd(8,2):  [1.5, [1.292893219, 4.548218497]]
# Gnd(8,3):  [3.,  [3.000000000, 8.485281374]]
# Gnd(9,2):  [1.5, [1.060307379, 4.118849118]]
# Gnd(9,3):  [3.,  [2.560307379, 7.838837739]]
# Gnd(10,2): [1.2, [0.881966012, 3.756521819]]
# Gnd(10,3): [2.4, [2.190983006, 7.251454484]]