URL='http://github.com/jasongrout/minimum_rank/raw/minimum_rank_1_1_3/'
files=['Zq_c.pyx','Zq.py','zero_forcing_64.pyx','zero_forcing_wavefront.pyx','minrank.py', 'inertia.py']
for f in files:
load(URL+f)
load('https://raw.githubusercontent.com/jephianlin/minimum_rank_aux/master/load_all.py');
Compiling /home/sageuser/9/.sage/temp/sage.math.iastate.edu/24299/tmp_PtSdHE.pyx..\ . Compiling /home/sageuser/9/.sage/temp/sage.math.iastate.edu/24299/tmp_dxnLDA.pyx..\ . Compiling /home/sageuser/9/.sage/temp/sage.math.iastate.edu/24299/tmp_GJbdY2.pyx..\ . Loading oc_diag_analysis.sage... ---gZ_leq, find_gZ, find_EZ, diagonal_analysis, etc. Loading xi_dict.py... ---SAPreduced_matrix, has_SAP, find_ZFloor, Zsap, etc. Loading mu_dict.py... ---get_mu_from_dict, has_minor, find_mu, etc. Loading general_Lib.py... ---sshow, empty_array, all_one_matrix, elementary_matrix, eigens_multi, sort_dictionary, etc. Compiling /home/sageuser/9/.sage/temp/sage.math.iastate.edu/24299/tmp_PtSdHE.pyx... Compiling /home/sageuser/9/.sage/temp/sage.math.iastate.edu/24299/tmp_dxnLDA.pyx... Compiling /home/sageuser/9/.sage/temp/sage.math.iastate.edu/24299/tmp_GJbdY2.pyx... Loading oc_diag_analysis.sage... ---gZ_leq, find_gZ, find_EZ, diagonal_analysis, etc. Loading xi_dict.py... ---SAPreduced_matrix, has_SAP, find_ZFloor, Zsap, etc. Loading mu_dict.py... ---get_mu_from_dict, has_minor, find_mu, etc. Loading general_Lib.py... ---sshow, empty_array, all_one_matrix, elementary_matrix, eigens_multi, sort_dictionary, etc. |
def L(k,l):
g=graphs.CompleteGraph(k);
for i in range(l):
g.add_vertex(k+i);
g.add_edge(k+i,k+i-1);
return g;
def lbd_Z(g):
n=g.order();
return integer_ceil((n-1)/(find_Z(g.complement())+1)+1);
def lbd_D(g):
n=g.order();
return integer_ceil((n-1)/(n-g.complement().diameter()+1)+1);
def lbd_girth(g):
n=g.order();
return integer_ceil((n-1)/(n-g.complement().girth()+2+1)+1);
def distinct_eigens(g):
return len(set(g.distance_matrix().eigenvalues()));
def all_one_matrix(m,n):
mn=m*n;
return matrix(m,[1]*mn);
def distance_decom(g):
n=g.order();
diam=g.diameter();
D=g.distance_matrix();
DD={};
for i in range(0,diam+1):
DD[i]=[];
for j in range(n):
DD[i].append([0]*n);
for i in range(n):
for j in range(n):
dist=D[i][j];
DD[dist][i][j]=1;
decom=[];
for i in range(0,diam+1):
decom.append(matrix(DD[i]));
return decom;
def commutative_family(mtxs):
## import as list;
n=len(mtxs);
commu=True;
for pair in Combinations(range(n),2):
if mtxs[pair[0]]*mtxs[pair[1]]!=mtxs[pair[1]]*mtxs[pair[0]]:
commu=False;
break;
return commu;
def is_CompleteBipartite(g):
h=g.complement();
com_sub=h.connected_components_subgraphs();
if len(com_sub)!=2:
return False;
for com in com_sub:
if min(com.degree())<com.order()-1:
return False;
return True;
def similar(A,Q):
## input A, Q and output Q^-1AQ
return Q.inverse()*A*Q;
def ttt(n):
l=[2]*n;
l[0]=3;
l[n-1]=3;
return l;
def caterpillar(n,l):
g=graphs.PathGraph(n);
for i in range(n):
for j in range(l[i]):
g.add_vertex((i,j));
g.add_edge(i,(i,j));
return g;
def find_valley(l):
##this code already assume l is unimodal, and it has just one valley;
n=len(l);
valley=min(l);
for i in range(n):
if l[i]==valley:
break;
return i;
def scale2(l):
n=len(l);
for k in range(n):
l[k]=l[k]*2^(k);
return l;
def bounds(n):
##upd=N(1-1/sqrt(5),digits=2);
return [N(float(n//2)/n),N(1-(1/sqrt(5)))];
def novertwo(n):
return [N(float(n//2)/n),N(float((n+1)//2)/n)];
def peak_ratio(g,d=2):
n=g.order();
D=g.distance_matrix();
p=D.characteristic_polynomial();
l=scale2(p.coeffs());
vly=find_valley(l);
ratio=N(float(vly)/n,digits=d);
return ratio;
def inertia(D):
egv=D.eigenvalues();
n=len(egv);
p=0;
m=0;
z=0;
for i in range(n):
if egv[i]<0:
m+=1;
if egv[i]>0:
p+=1;
return [p,m,z];
def eigens_multi(A):
l=A.eigenvalues();
eigens={};
for i in l:
eigens[i]=0;
for i in l:
eigens[i]+=1;
return eigens;
def sort_dictionary(d):
l=d.keys();
l.sort();
#print l;
for key in l:
print "%s:%s"%(key,d[key]);
#print " ";
def KneserPartition(n,k):
## look at the Knesergraph K(n,k)
## output the list of [C0, C1, ..., Ck];
## let alpha = range(1,k)
## Ci is the "list" of vertex beta such that |alpha\cap beta|=k-i;
## C0 is the just [alpha];
g=graphs.KneserGraph(n,k);
C=[0]*(k+1);
for i in range(k+1):
C[i]=[];
alpha=EnumeratedSet(range(1,k+1));
for v in g.vertices():
inter=len(alpha.intersection(v));
i=k-inter;
#print i;
C[i].append(v);
return C;
def quotient_distance_matrix(g,C):
Q=[];
l=len(C);
for i in range(l):
Q.append([0]*(l));
for i in range(l):
for j in range(l):
row_sum=0;
a=C[i][0]; #a is an element in C[i]
for c in C[j]: #for every c in C[j]
#print i,j,a,c,row_sum;
row_sum+=g.distance(a,c);
Q[i][j]=row_sum;
return matrix(Q);
def quotient_distance_decom(g,C):
diam=g.diameter();
Q=[0]*(diam+1);
for i in range(diam+1):
Q[i]=[];
l=len(C);
for i in range(diam+1):
for j in range(l):
Q[i].append([0]*(l));
for i in range(l):
for j in range(l):
row_sum=[0]*(diam+1);
a=C[i][0]; #a is an element in C[i]
for c in C[j]: #for every c in C[j]
#print i,j,a,c,row_sum;
row_sum[g.distance(a,c)]+=1;
for k in range(diam+1):
Q[k][i][j]=row_sum[k];
for k in range(diam+1):
Q[k]=matrix(Q[k]);
return Q;
def novertwo(n):
return [N(float(n//2)/n),N(float((n+1)//2)/n)];
def double_star(a,b,c):
#return a double star with a leaves on one side and b leaves on the other;
#the length of the middle path is c-1;
g=graphs.PathGraph(c);
for i in range(a):
g.add_vertex(-i-1);
g.add_edge(0,-i-1);
for i in range(b):
g.add_vertex(c+i);
g.add_edge(c-1,c+i);
return g;
def broom_partition(a,c):
C=[];
C1=[];
for i in range(a):
C1.append(-i-1);
C.append(C1);
for i in range(c):
C.append([i]);
return C;
|
|
g=Graph("CF")
show(g);
p=g.distance_matrix().characteristic_polynomial();
find_valley(scale2(p.coeffs()))
print scale2(p.coeffs())
 [-12, -56, -60, 0, 16] [-12, -56, -60, 0, 16] |
print "This is for paths on n vertices."
for n in range(1,31):
g=graphs.PathGraph(n);
print n, peak_ratio(g,d=6)*n,bounds(n)[1]*n;
This is for paths on n vertices. 1 0.000000 0.552786404500042 2 0.000000 1.10557280900008 3 1.00000 1.65835921350013 4 2.00000 2.21114561800017 5 2.00000 2.76393202250021 6 3.00000 3.31671842700025 7 3.00000 3.86950483150029 8 4.00000 4.42229123600034 9 5.00000 4.97507764050038 10 5.00000 5.52786404500042 11 6.00000 6.08065044950046 12 6.00000 6.63343685400050 13 7.00000 7.18622325850055 14 7.00000 7.73900966300059 15 8.00000 8.29179606750063 16 8.00000 8.84458247200067 17 9.00000 9.39736887650071 18 10.0000 9.95015528100076 19 10.0000 10.5029416855008 20 11.0000 11.0557280900008 21 11.0000 11.6085144945009 22 12.0000 12.1613008990009 23 12.0000 12.7140873035010 24 13.0000 13.2668737080010 25 13.0000 13.8196601125011 26 14.0000 14.3724465170011 27 15.0000 14.9252329215011 28 15.0000 15.4780193260012 29 16.0000 16.0308057305012 30 16.0000 16.5835921350013 This is for paths on n vertices. 1 0.000000 0.552786404500042 2 0.000000 1.10557280900008 3 1.00000 1.65835921350013 4 2.00000 2.21114561800017 5 2.00000 2.76393202250021 6 3.00000 3.31671842700025 7 3.00000 3.86950483150029 8 4.00000 4.42229123600034 9 5.00000 4.97507764050038 10 5.00000 5.52786404500042 11 6.00000 6.08065044950046 12 6.00000 6.63343685400050 13 7.00000 7.18622325850055 14 7.00000 7.73900966300059 15 8.00000 8.29179606750063 16 8.00000 8.84458247200067 17 9.00000 9.39736887650071 18 10.0000 9.95015528100076 19 10.0000 10.5029416855008 20 11.0000 11.0557280900008 21 11.0000 11.6085144945009 22 12.0000 12.1613008990009 23 12.0000 12.7140873035010 24 13.0000 13.2668737080010 25 13.0000 13.8196601125011 26 14.0000 14.3724465170011 27 15.0000 14.9252329215011 28 15.0000 15.4780193260012 29 16.0000 16.0308057305012 30 16.0000 16.5835921350013 |
print "This is for brooms on n vertices with two leaves on one end."
for n in range(3,31):
g=double_star(2,0,n-2);
print n, peak_ratio(g,d=6), bounds(n)[1];
#sshow(g);
This is for brooms on n vertices with two leaves on one end. 3 0.333333 0.552786404500042 4 0.500000 0.552786404500042 5 0.400000 0.552786404500042 6 0.500000 0.552786404500042 7 0.428571 0.552786404500042 8 0.500000 0.552786404500042 9 0.555556 0.552786404500042 10 0.500000 0.552786404500042 11 0.545455 0.552786404500042 12 0.500000 0.552786404500042 13 0.538462 0.552786404500042 14 0.500000 0.552786404500042 15 0.533333 0.552786404500042 16 0.500000 0.552786404500042 17 0.529412 0.552786404500042 18 0.555556 0.552786404500042 19 0.526316 0.552786404500042 20 0.550000 0.552786404500042 21 0.523810 0.552786404500042 22 0.545455 0.552786404500042 23 0.521739 0.552786404500042 24 0.541667 0.552786404500042 25 0.520000 0.552786404500042 26 0.538462 0.552786404500042 27 0.555556 0.552786404500042 28 0.535714 0.552786404500042 29 0.551724 0.552786404500042 30 0.533333 0.552786404500042 This is for brooms on n vertices with two leaves on one end. 3 0.333333 0.552786404500042 4 0.500000 0.552786404500042 5 0.400000 0.552786404500042 6 0.500000 0.552786404500042 7 0.428571 0.552786404500042 8 0.500000 0.552786404500042 9 0.555556 0.552786404500042 10 0.500000 0.552786404500042 11 0.545455 0.552786404500042 12 0.500000 0.552786404500042 13 0.538462 0.552786404500042 14 0.500000 0.552786404500042 15 0.533333 0.552786404500042 16 0.500000 0.552786404500042 17 0.529412 0.552786404500042 18 0.555556 0.552786404500042 19 0.526316 0.552786404500042 20 0.550000 0.552786404500042 21 0.523810 0.552786404500042 22 0.545455 0.552786404500042 23 0.521739 0.552786404500042 24 0.541667 0.552786404500042 25 0.520000 0.552786404500042 26 0.538462 0.552786404500042 27 0.555556 0.552786404500042 28 0.535714 0.552786404500042 29 0.551724 0.552786404500042 30 0.533333 0.552786404500042 |
for n in range(450,451):
g=graphs.PathGraph(n);
print n, peak_ratio(g,d=6),bounds(n)
450 0.551111 [0.500000000000000, 0.552786404500042] 450 0.551111 [0.500000000000000, 0.552786404500042] |
a,c=5,2;
h=double_star(a,0,c);
show(h)
C=broom_partition(a,c);
#print C;
D=h.distance_matrix();
Dq=quotient_distance_matrix(h,C);
show(D);
show(Dq);
sort_dictionary(eigens_multi(D))
print "---"
sort_dictionary(eigens_multi(Dq))
f1=D.characteristic_polynomial();
f2=Dq.characteristic_polynomial();
show(f1);
show(f2);
x=var("x");
show(expand(f2*(x+2)^(a-1)));
|
|
### Need to check if peak_ratio>=n/2 ceil => diam(t)
### two graphs found on n=22: U???????????????????K?AO?O_?FF?G?[G??{?? U???????????????????c?AO?C_B@??w?M@?{?_?
for n in range(1,31):
upd=novertwo(n)[1];
for g in graphs.nauty_geng("%s %s:%s -c"%(n,n-1,n-1)):
if g.diameter()<n/2:
if peak_ratio(g)>upd:
print n, g.canonical_label().graph6_string();
print n,"done";
|
|
c=-1;
for a in range(1,8):
g=double_star(a,0,c);
C=broom_partition(a,c);
#print C;
D=g.distance_matrix();
Dq=quotient_distance_matrix(g,C);
#show(Dq)
print a;
show(Dq.characteristic_polynomial())
#print eigens_multi(Dq);
1 2 3 4 5 6 7 1 2 3 4 5 6 7 |
c=1;
for a in range(1,8):
g=double_star(a,0,c);
C=broom_partition(a,c);
#print C;
D=g.distance_matrix();
Dq=quotient_distance_matrix(g,C);
#show(Dq)
print a;
show(D.characteristic_polynomial())
#print eigens_multi(Dq);
1 2 3 4 5 6 7 1 2 3 4 5 6 7 |
g=double_star(4,0,1);
D=g.distance_matrix();
show(D.characteristic_polynomial());
|
|
n,counter=199,1;
while True:
g=graphs.RandomTree(n);
p=g.distance_matrix().characteristic_polynomial();
vly=find_valley(scale2(p.coeffs()));
if vly<n//2 or vly>integer_ceil(n*(1-1/sqrt(5))):
print n,g.graph6_string();
break;
if counter%100==0:
print counter;
counter+=1;
WARNING: Output truncated!full_output.txt 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 ... 69700 69800 69900 70000 70100 70200 70300 70400 70500 70600 70700 70800 70900 71000 71100 71200 71300 71400 71500 71600 71700 71800 71900 72000 72100 72200 72300 72400 72500 72600 72700 72800 72900 73000 73100 73200 73300 73400 73500 73600 73700 73800 73900 74000 74100 74200 74300 74400 74500 74600 74700 74800 74900 75000 75100 75200 75300 75400 75500 you are running out of primes. 1000 coprime primes found199 ~?BF????????????????????????G????????????????????????????_?????????A????\ ??????????????????????????????G??????????????C????A??????_??????????????\ ????A????????????G?C????????C??????????????????????????????????????_????\ ?????????????????????_???_????@????????????????????C????????????????????\ ???????@????????????????????????????????????????????????????????????????\ ??????????????a??????????????_?????????????????????????????@????????????\ ?????????????????????@???????????A??????????????????????????????A???????\ ??????@?A????????????????_??????????????????????????????????????O???????\ ????????????????????_???????????G??_?????????????????????_??????????????\ ????????C_?????????A??????????????????????????A????O@???????????????????\ ??????????????????S????????????????????????????????????????????????O????\ ????@??????????????????_??????????????????????????O?????????????????????\ ??????????????????????????C?????????????????????????_???????????????????\ ?????O????????????G???????????????????????A?@???????????????????????????\ ??????O?????????????_????????_???@????????????????????O??????????????C??\ ????O?????@???????@?????????????????@?????????????????????@??????@??????\ ???G??????????????????@?????????????????????@???????????????????????????\ ?????G???????G?????G?????????????????????????????A??G?????????_???@?????\ ???????????????????@??????G????????O?????????????????????@??????????????\ ??????????????????????O?????P????????_????????????C?????????????_???????\ ????????????????O??????O??????????????????@?????????????@???????????????\ ???????????????C???????????????????????C?????????G?????????A????????????\ ?????_?????????O????????@????????G????????????A????C??????????C?????????\ ???????CA??????????A??????????????????????????????????C???????????A?????\ ?????@?????????A???????@????O???????????????????A@?????C????????????????\ ????????????????????C???A??????C???????????????????????????_????????????\ ?A?????O???????????????????????G???????????????????????????????????_????\ ??C???A????????????????????????????_???????????????????????A????????????\ ?A??????????????????????C???????????????????????????????????@???G???????\ ?????????G?????O????????????????????_????????????C??????????????????????\ ???????????????C???????????C??????_???@?O????????????????C??????????????\ O??????????????????A????????????????C????????????????????????E??????????\ G???????????????????????????G????????????????????C???G??????????????????\ ??????C??O?????????????????G??@??????????????????????????????O??????????\ ????????????????????????????????????A????????????????????????O??????@???\ ????????????????@???????????????????????????_?????????W?????????????????\ ?????????????????????????@??????????????????????????????????????????????\ ????_????????????@????????????G???????????????G???????????O?????????????\ ??????????????_?CO???????@?????????????????????????????????????????????O\ ??@?????????????????????????????????????A??_????????????????????????????\ ???C???@???C?????????????C???????????AC??????????????????????????????_??\ ????????????G???@A??????????????????????????A?????????????????C???????GG\ ????????????????????G????????????????C??????_???????????????????A???????\ ??????????_A?????????????????????????????????????????A????????????????_?\ ?????????????????_???????????????????????????A??????????_???????????????\ ??????O????G??????????O????????????????????????? WARNING: Output truncated!full_output.txt 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 ... 69700 69800 69900 70000 70100 70200 70300 70400 70500 70600 70700 70800 70900 71000 71100 71200 71300 71400 71500 71600 71700 71800 71900 72000 72100 72200 72300 72400 72500 72600 72700 72800 72900 73000 73100 73200 73300 73400 73500 73600 73700 73800 73900 74000 74100 74200 74300 74400 74500 74600 74700 74800 74900 75000 75100 75200 75300 75400 75500 you are running out of primes. 1000 coprime primes found199 ~?BF????????????????????????G????????????????????????????_?????????A??????????????????????????????????G??????????????C????A??????_??????????????????A????????????G?C????????C??????????????????????????????????????_?????????????????????????_???_????@????????????????????C???????????????????????????@??????????????????????????????????????????????????????????????????????????????a??????????????_?????????????????????????????@?????????????????????????????????@???????????A??????????????????????????????A?????????????@?A????????????????_??????????????????????????????????????O???????????????????????????_???????????G??_?????????????????????_??????????????????????C_?????????A??????????????????????????A????O@?????????????????????????????????????S????????????????????????????????????????????????O????????@??????????????????_??????????????????????????O???????????????????????????????????????????????C?????????????????????????_????????????????????????O????????????G???????????????????????A?@?????????????????????????????????O?????????????_????????_???@????????????????????O??????????????C??????O?????@???????@?????????????????@?????????????????????@??????@?????????G??????????????????@?????????????????????@????????????????????????????????G???????G?????G?????????????????????????????A??G?????????_???@????????????????????????@??????G????????O?????????????????????@????????????????????????????????????O?????P????????_????????????C?????????????_???????????????????????O??????O??????????????????@?????????????@??????????????????????????????C???????????????????????C?????????G?????????A?????????????????_?????????O????????@????????G????????????A????C??????????C????????????????CA??????????A??????????????????????????????????C???????????A??????????@?????????A???????@????O???????????????????A@?????C????????????????????????????????????C???A??????C???????????????????????????_?????????????A?????O???????????????????????G???????????????????????????????????_??????C???A????????????????????????????_???????????????????????A?????????????A??????????????????????C???????????????????????????????????@???G????????????????G?????O????????????????????_????????????C?????????????????????????????????????C???????????C??????_???@?O????????????????C??????????????O??????????????????A????????????????C????????????????????????E??????????G???????????????????????????G????????????????????C???G????????????????????????C??O?????????????????G??@??????????????????????????????O??????????????????????????????????????????????A????????????????????????O??????@???????????????????@???????????????????????????_?????????W??????????????????????????????????????????@??????????????????????????????????????????????????_????????????@????????????G???????????????G???????????O???????????????????????????_?CO???????@?????????????????????????????????????????????O??@?????????????????????????????????????A??_???????????????????????????????C???@???C?????????????C???????????AC??????????????????????????????_??????????????G???@A??????????????????????????A?????????????????C???????GG????????????????????G????????????????C??????_???????????????????A?????????????????_A?????????????????????????????????????????A????????????????_??????????????????_???????????????????????????A??????????_?????????????????????O????G??????????O????????????????????????? |
g=Graph("~?BF????????????????????????G????????????????????????????_?????????A????\
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os.system("sage -i nauty-25")
tee: /home/sage/build/sage-6.2/logs/install.log: Permission denied tee: /home/sage/build/sage-6.2/logs/pkgs/nauty-25.log: Permission denied Found package nauty-25 in /home/sage/build/sage-6.2/upstream/nauty-25.spkg Package nauty-25 is already installed. Use 'sage -f /home/sage/build/sage-6.2/upstream/nauty-25.spkg' to force a reinstallation. touch: cannot touch `/home/sage/build/sage-6.2/local/var/lib/sage/installed/nauty-25': Permission denied 256 tee: /home/sage/build/sage-6.2/logs/install.log: Permission denied tee: /home/sage/build/sage-6.2/logs/pkgs/nauty-25.log: Permission denied Found package nauty-25 in /home/sage/build/sage-6.2/upstream/nauty-25.spkg Package nauty-25 is already installed. Use 'sage -f /home/sage/build/sage-6.2/upstream/nauty-25.spkg' to force a reinstallation. touch: cannot touch `/home/sage/build/sage-6.2/local/var/lib/sage/installed/nauty-25': Permission denied 256 |
for n in range(4):
print n;
for g in graphs.nauty_geng("%s -c"%n):
print g.graph6_string();
reset();
0 1 @ 2 A_ 3 BW Bw 0 1 @ 2 A_ 3 BW Bw |
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