$ ./flagmatic --n 7 --r 3 --induced-density 4.2 --forbid-k4- --forbid-f32 --verbose --max-flags 150 --dir output/k4-f32max42 flagmatic version 1.5 ============================================================================ Optimizing for density of 4.2. Using directory output/k4-f32max42 Forbidding 4.3 Forbidding 5:123124125345 Using admissible graphs of order 7. Generating types and flags... Generated 1 type of order 1, with 5 flags of order 4. Generated 2 types of order 3, with [92, 43] flags of order 5. Generated 10 types of order 5, with [388, 245, 169, 157, 142, 111, 111, 75, 78, 57] flags of order 6. 4 types removed; remaining types have [5, 92, 43, 142, 111, 111, 75, 78, 57] flags. Generating admissible graphs... Generated 5173 admissible graphs. Written flags.py Computing flag densities... Written flags.dat-s and flags.rat Running: ./csdp output/k4-f32max42/flags.dat-s output/k4-f32max42/flags.out ============================================================================ Iter: 0 Ap: 0.00e+00 Pobj: -4.4015060e+04 Ad: 0.00e+00 Dobj: 0.0000000e+00 Iter: 1 Ap: 1.00e+00 Pobj: -5.2910665e+04 Ad: 9.53e-01 Dobj: -2.8150813e-01 Iter: 2 Ap: 1.00e+00 Pobj: -5.8217192e+04 Ad: 8.48e-01 Dobj: -2.8424901e-01 Iter: 3 Ap: 1.00e+00 Pobj: -8.1825117e+04 Ad: 4.17e-01 Dobj: -2.8448780e-01 Iter: 4 Ap: 2.78e-01 Pobj: -8.2929630e+04 Ad: 3.40e-01 Dobj: -2.6921658e-01 Iter: 5 Ap: 2.05e-01 Pobj: -8.6347347e+04 Ad: 4.97e-01 Dobj: -2.7369561e-01 Iter: 6 Ap: 5.58e-01 Pobj: -8.9020156e+04 Ad: 3.41e-01 Dobj: -2.6618918e-01 Iter: 7 Ap: 5.82e-01 Pobj: -9.7643223e+04 Ad: 4.38e-01 Dobj: -2.6134795e-01 Iter: 8 Ap: 3.34e-01 Pobj: -9.8943824e+04 Ad: 4.75e-01 Dobj: -2.5612974e-01 Iter: 9 Ap: 5.18e-01 Pobj: -9.3047698e+04 Ad: 5.09e-01 Dobj: -2.5404733e-01 Iter: 10 Ap: 7.38e-01 Pobj: -7.7657369e+04 Ad: 5.50e-01 Dobj: -2.5159997e-01 Iter: 11 Ap: 8.43e-01 Pobj: -5.6774740e+04 Ad: 6.23e-01 Dobj: -2.5059322e-01 Iter: 12 Ap: 3.99e-01 Pobj: -4.8998929e+04 Ad: 5.64e-01 Dobj: -2.5061467e-01 Iter: 13 Ap: 6.34e-01 Pobj: -3.4256857e+04 Ad: 5.65e-01 Dobj: -2.5067342e-01 Iter: 14 Ap: 2.95e-01 Pobj: -2.9519389e+04 Ad: 5.22e-01 Dobj: -2.5069324e-01 Iter: 15 Ap: 4.98e-01 Pobj: -2.1835679e+04 Ad: 4.21e-01 Dobj: -2.5090121e-01 Iter: 16 Ap: 3.51e-01 Pobj: -1.8835650e+04 Ad: 5.71e-01 Dobj: -2.5125222e-01 Iter: 17 Ap: 1.78e-01 Pobj: -1.7321527e+04 Ad: 4.94e-01 Dobj: -2.5161713e-01 Iter: 18 Ap: 4.54e-01 Pobj: -1.4485800e+04 Ad: 4.76e-01 Dobj: -2.5209134e-01 Iter: 19 Ap: 6.65e-01 Pobj: -1.0845797e+04 Ad: 7.22e-01 Dobj: -2.5314608e-01 Iter: 20 Ap: 9.82e-01 Pobj: -4.5980487e+03 Ad: 1.00e+00 Dobj: -2.5378814e-01 Iter: 21 Ap: 9.72e-01 Pobj: -3.1413739e+02 Ad: 1.00e+00 Dobj: -2.5382575e-01 Iter: 22 Ap: 9.94e-01 Pobj: -1.6689978e+01 Ad: 1.00e+00 Dobj: -2.5392360e-01 Iter: 23 Ap: 1.00e+00 Pobj: -2.4203015e+00 Ad: 1.00e+00 Dobj: -2.5522534e-01 Iter: 24 Ap: 1.00e+00 Pobj: -1.4144104e+00 Ad: 1.00e+00 Dobj: -2.6493872e-01 Iter: 25 Ap: 4.70e-01 Pobj: -1.2205016e+00 Ad: 1.00e+00 Dobj: -2.7330254e-01 Iter: 26 Ap: 1.00e+00 Pobj: -9.0034809e-01 Ad: 1.00e+00 Dobj: -2.8471008e-01 Iter: 27 Ap: 1.00e+00 Pobj: -6.6998214e-01 Ad: 1.00e+00 Dobj: -3.0359952e-01 Iter: 28 Ap: 4.62e-01 Pobj: -6.6388063e-01 Ad: 6.56e-01 Dobj: -3.3947732e-01 Iter: 29 Ap: 9.26e-02 Pobj: -6.6107681e-01 Ad: 4.43e-01 Dobj: -3.5358906e-01 Iter: 30 Ap: 7.08e-02 Pobj: -6.6060367e-01 Ad: 3.86e-01 Dobj: -3.6738303e-01 Iter: 31 Ap: 3.82e-01 Pobj: -6.5230441e-01 Ad: 8.42e-01 Dobj: -4.1059855e-01 Iter: 32 Ap: 5.67e-01 Pobj: -6.2381070e-01 Ad: 1.00e+00 Dobj: -4.5246709e-01 Iter: 33 Ap: 9.55e-01 Pobj: -5.7940205e-01 Ad: 1.00e+00 Dobj: -4.9317320e-01 Iter: 34 Ap: 1.00e+00 Pobj: -5.6039608e-01 Ad: 1.00e+00 Dobj: -5.2590993e-01 Iter: 35 Ap: 8.05e-01 Pobj: -5.5690551e-01 Ad: 1.00e+00 Dobj: -5.3996238e-01 Iter: 36 Ap: 1.00e+00 Pobj: -5.5563850e-01 Ad: 1.00e+00 Dobj: -5.4872668e-01 Iter: 37 Ap: 8.41e-01 Pobj: -5.5558050e-01 Ad: 1.00e+00 Dobj: -5.5284250e-01 Iter: 38 Ap: 7.58e-01 Pobj: -5.5556698e-01 Ad: 1.00e+00 Dobj: -5.5411008e-01 Iter: 39 Ap: 7.05e-01 Pobj: -5.5556172e-01 Ad: 1.00e+00 Dobj: -5.5477240e-01 Iter: 40 Ap: 8.15e-01 Pobj: -5.5555859e-01 Ad: 1.00e+00 Dobj: -5.5512688e-01 Iter: 41 Ap: 8.47e-01 Pobj: -5.5555713e-01 Ad: 1.00e+00 Dobj: -5.5534211e-01 Iter: 42 Ap: 7.62e-01 Pobj: -5.5555653e-01 Ad: 1.00e+00 Dobj: -5.5544424e-01 Iter: 43 Ap: 5.34e-01 Pobj: -5.5555632e-01 Ad: 1.00e+00 Dobj: -5.5548157e-01 Iter: 44 Ap: 5.12e-01 Pobj: -5.5555616e-01 Ad: 1.00e+00 Dobj: -5.5550448e-01 Iter: 45 Ap: 8.17e-01 Pobj: -5.5555592e-01 Ad: 1.00e+00 Dobj: -5.5552788e-01 Iter: 46 Ap: 6.73e-01 Pobj: -5.5555583e-01 Ad: 9.60e-01 Dobj: -5.5553931e-01 Iter: 47 Ap: 7.13e-01 Pobj: -5.5555575e-01 Ad: 1.00e+00 Dobj: -5.5554881e-01 Iter: 48 Ap: 3.80e-01 Pobj: -5.5555573e-01 Ad: 7.78e-01 Dobj: -5.5555094e-01 Iter: 49 Ap: 8.28e-01 Pobj: -5.5555567e-01 Ad: 1.00e+00 Dobj: -5.5555498e-01 Iter: 50 Ap: 5.48e-01 Pobj: -5.5555564e-01 Ad: 9.74e-01 Dobj: -5.5555831e-01 Iter: 51 Ap: 7.48e-01 Pobj: -5.5555561e-01 Ad: 1.00e+00 Dobj: -5.5555806e-01 Iter: 52 Ap: 1.00e+00 Pobj: -5.5555557e-01 Ad: 1.00e+00 Dobj: -5.5555726e-01 Iter: 53 Ap: 1.00e+00 Pobj: -5.5555556e-01 Ad: 9.92e-01 Dobj: -5.5555587e-01 Iter: 54 Ap: 1.00e+00 Pobj: -5.5555556e-01 Ad: 9.41e-01 Dobj: -5.5555559e-01 Iter: 55 Ap: 9.47e-01 Pobj: -5.5555556e-01 Ad: 9.42e-01 Dobj: -5.5555556e-01 Success: SDP solved Primal objective value: -5.5555556e-01 Dual objective value: -5.5555556e-01 Relative primal infeasibility: 3.07e-11 Relative dual infeasibility: 3.49e-10 Real Relative Gap: -1.83e-09 XZ Relative Gap: 3.02e-09 DIMACS error measures: 3.78e-10 0.00e+00 6.80e-10 0.00e+00 -1.83e-09 3.02e-09 Elements time: 2819.495000 Factor time: 86.849948 Other time: 47.394806 Total time: 2953.739754 ============================================================================ Return code is 0 Approximate floating-point bound is 0.55555556 $ sage -python scripts/find_sharp_graphs.py --dir output/k4-f32max42 Floating point bound is 0.555555555571330850. 9 members of H are sharp. 0.555555555563137626 : graph 1 (7:) 0.555555555564409165 : graph 46 (7:123124125126127) 0.555555555571330850 : graph 1066 (7:123124125126137147157167) 0.555555555566501713 : graph 2209 (7:123124125136137146147156157) 0.555555555566084602 : graph 4556 (7:123124125126137147157267367467567) 0.555555555568612580 : graph 5022 (7:123124125126137147257267357367457467) 0.555555555568425508 : graph 5042 (7:123124125136146156237247257367467567) 0.555555555567213255 : graph 5143 (7:123124125136137146147256257356357456457) 0.555555555559955283 : graph 5173 (7:123124125136146157167237247256267356357456457) Written sharp graphs to flags.py $ sage -python scripts/check_construction.py --n 7 --r 3 --induced-density 4.2 --vertex-transitive 6:123234345451512136246356256146 Density of 4.2 is 5/9. 9 graphs of order 7 occur as induced subgraphs of the blow-up: 7: has density 1663/23328 (0.071288) 7:123124125126127 has density 35/486 (0.072016) 7:123124125126137147157167 has density 175/864 (0.202546) 7:123124125136137146147156157 has density 1225/11664 (0.105024) 7:123124125126137147157267367467567 has density 175/1944 (0.090021) 7:123124125136146156237247257367467567 has density 175/1296 (0.135031) 7:123124125126137147257267357367457467 has density 175/1296 (0.135031) 7:123124125136137146147256257356357456457 has density 175/1296 (0.135031) 7:123124125136146157167237247256267356357456457 has density 35/648 (0.054012) $ sage -python scripts/make_zero_eigenvectors.py --vertex-transitive 6:123234345451512136246356256146 --dir output/k4-f32max42 Constructed 1 out of 1 zero eigenvectors for type 1. Constructed 5 out of 5 zero eigenvectors for type 2. Constructed 1 out of 1 zero eigenvectors for type 3. Constructed 4 out of 4 zero eigenvectors for type 4. Constructed 0 out of 0 zero eigenvectors for type 5. Constructed 0 out of 0 zero eigenvectors for type 6. Constructed 3 out of 3 zero eigenvectors for type 7. Constructed 0 out of 0 zero eigenvectors for type 8. Constructed 1 out of 1 zero eigenvectors for type 9. Written zev.py Written field to flags.py $ sage -python scripts/factor_approximate_q.py --dir output/k4-f32max42 Floating point bound is 0.555555555571330850. Type 1: smallest eigenvalue is 0.008615329658495185 Type 2: smallest eigenvalue is 0.029157645321189113 Type 3: smallest eigenvalue is 0.019957861257191574 Type 4: smallest eigenvalue is 0.098859560608587249 Type 5: smallest eigenvalue is 0.090030702496723347 Type 6: smallest eigenvalue is 0.108338064260138547 Type 7: smallest eigenvalue is 0.093855517452594781 Type 8: smallest eigenvalue is 0.082478713253693242 Type 9: smallest eigenvalue is 0.053084161366263136 Written r.py Written qdashf.py $ sage -python scripts/make_exact_qdash.py '5/9' --denominator 240 --dir output/k4-f32max42 --diagonalize Type 1: smallest eigenvalue is 0.009419822288592072 Type 2: smallest eigenvalue is 0.019158034367165268 Type 3: smallest eigenvalue is 0.018479529698709484 Type 4: smallest eigenvalue is 0.053455014840929092 Type 5: smallest eigenvalue is 0.090649810253679394 Type 6: smallest eigenvalue is 0.109924858380208754 Type 7: smallest eigenvalue is 0.092776386239041331 Type 8: smallest eigenvalue is 0.082943158522163135 Type 9: smallest eigenvalue is 0.051648050847463489 Diagonalizing matrices... Written qdash.py Written r.py Added exact bound to flags.py $ sage -python scripts/verify_bound.py --dir output/k4-f32max42 Written q.py Floating point bound (non-sharp graphs) is 0.555482511601510409 Exact bound (just sharp graphs) is 5/9 Bound (all graphs) is 5/9 $ sage -python scripts/make_certificate.py --dir output/k4-f32max42 Written certificate to cert.js