$ ./flagmatic --r 3 --n 7 --forbid-k4- --forbid-f32 --forbid-c5 --verbose --max-flags 150 --dir output/k4-f32c5 flagmatic version 1.5 ============================================================================ Using directory output/k4-f32c5 Forbidding 4.3 Forbidding 5:123124135245345 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 [89, 40] flags of order 5. Generated 9 types of order 5, with [388, 245, 153, 157, 106, 98, 103, 63, 66] flags of order 6. 4 types removed; remaining types have [5, 89, 40, 106, 98, 103, 63, 66] flags. Generating admissible graphs... Generated 4045 admissible graphs. Written flags.py Computing flag densities... Written flags.dat-s and flags.rat Running: ./csdp output/k4-f32c5/flags.dat-s output/k4-f32c5/flags.out ============================================================================ Iter: 0 Ap: 0.00e+00 Pobj: -2.5634168e+04 Ad: 0.00e+00 Dobj: 0.0000000e+00 Iter: 1 Ap: 1.00e+00 Pobj: -3.1153159e+04 Ad: 9.54e-01 Dobj: -1.9309361e-01 Iter: 2 Ap: 1.00e+00 Pobj: -3.4841376e+04 Ad: 8.29e-01 Dobj: -2.3184719e-01 Iter: 3 Ap: 1.00e+00 Pobj: -4.7604966e+04 Ad: 4.56e-01 Dobj: -2.3341740e-01 Iter: 4 Ap: 3.32e-01 Pobj: -4.8179739e+04 Ad: 3.52e-01 Dobj: -2.2016386e-01 Iter: 5 Ap: 2.04e-01 Pobj: -4.8583421e+04 Ad: 4.69e-01 Dobj: -2.0652857e-01 Iter: 6 Ap: 1.87e-01 Pobj: -4.9352896e+04 Ad: 3.77e-01 Dobj: -2.0002270e-01 Iter: 7 Ap: 3.43e-01 Pobj: -4.9052394e+04 Ad: 4.47e-01 Dobj: -1.9432629e-01 Iter: 8 Ap: 7.25e-01 Pobj: -4.4252308e+04 Ad: 5.21e-01 Dobj: -1.8963763e-01 Iter: 9 Ap: 4.36e-01 Pobj: -4.2610443e+04 Ad: 4.05e-01 Dobj: -1.8732283e-01 Iter: 10 Ap: 3.80e-01 Pobj: -4.0775294e+04 Ad: 5.43e-01 Dobj: -1.8592245e-01 Iter: 11 Ap: 5.50e-01 Pobj: -3.3788839e+04 Ad: 4.88e-01 Dobj: -1.8504989e-01 Iter: 12 Ap: 8.06e-01 Pobj: -2.6692967e+04 Ad: 6.06e-01 Dobj: -1.8486295e-01 Iter: 13 Ap: 6.24e-01 Pobj: -2.0652471e+04 Ad: 5.62e-01 Dobj: -1.8463629e-01 Iter: 14 Ap: 4.88e-01 Pobj: -1.7388256e+04 Ad: 5.15e-01 Dobj: -1.8435024e-01 Iter: 15 Ap: 4.90e-01 Pobj: -1.4484797e+04 Ad: 6.21e-01 Dobj: -1.8423988e-01 Iter: 16 Ap: 1.18e-01 Pobj: -1.3907842e+04 Ad: 4.08e-01 Dobj: -1.8417572e-01 Iter: 17 Ap: 5.09e-01 Pobj: -1.1735454e+04 Ad: 4.44e-01 Dobj: -1.8419908e-01 Iter: 18 Ap: 5.88e-01 Pobj: -9.0905371e+03 Ad: 6.79e-01 Dobj: -1.8427439e-01 Iter: 19 Ap: 8.81e-01 Pobj: -4.5564213e+03 Ad: 9.32e-01 Dobj: -1.8438847e-01 Iter: 20 Ap: 1.00e+00 Pobj: -8.8186811e+02 Ad: 1.00e+00 Dobj: -1.8442846e-01 Iter: 21 Ap: 9.76e-01 Pobj: -5.6025147e+01 Ad: 1.00e+00 Dobj: -1.8443615e-01 Iter: 22 Ap: 1.00e+00 Pobj: -2.9448332e+00 Ad: 1.00e+00 Dobj: -1.8448335e-01 Iter: 23 Ap: 1.00e+00 Pobj: -7.0977034e-01 Ad: 1.00e+00 Dobj: -1.8533131e-01 Iter: 24 Ap: 1.00e+00 Pobj: -4.8766385e-01 Ad: 8.98e-01 Dobj: -1.8905834e-01 Iter: 25 Ap: 2.87e-01 Pobj: -4.6178639e-01 Ad: 1.00e+00 Dobj: -1.8882427e-01 Iter: 26 Ap: 1.00e+00 Pobj: -3.6915310e-01 Ad: 8.18e-01 Dobj: -1.9290368e-01 Iter: 27 Ap: 6.58e-01 Pobj: -3.3934033e-01 Ad: 1.00e+00 Dobj: -1.9590345e-01 Iter: 28 Ap: 1.00e+00 Pobj: -2.8561913e-01 Ad: 1.00e+00 Dobj: -2.0150896e-01 Iter: 29 Ap: 6.20e-01 Pobj: -2.7372794e-01 Ad: 1.00e+00 Dobj: -2.0583931e-01 Iter: 30 Ap: 9.25e-01 Pobj: -2.6082685e-01 Ad: 1.00e+00 Dobj: -2.1461657e-01 Iter: 31 Ap: 1.51e-01 Pobj: -2.5989789e-01 Ad: 7.18e-01 Dobj: -2.1808382e-01 Iter: 32 Ap: 1.47e-01 Pobj: -2.5935361e-01 Ad: 9.29e-01 Dobj: -2.2446333e-01 Iter: 33 Ap: 6.02e-01 Pobj: -2.5506870e-01 Ad: 1.00e+00 Dobj: -2.2913941e-01 Iter: 34 Ap: 1.00e+00 Pobj: -2.4799648e-01 Ad: 1.00e+00 Dobj: -2.3592955e-01 Iter: 35 Ap: 9.21e-01 Pobj: -2.4556737e-01 Ad: 1.00e+00 Dobj: -2.4161108e-01 Iter: 36 Ap: 9.12e-01 Pobj: -2.4502442e-01 Ad: 1.00e+00 Dobj: -2.4272746e-01 Iter: 37 Ap: 1.00e+00 Pobj: -2.4491273e-01 Ad: 1.00e+00 Dobj: -2.4400865e-01 Iter: 38 Ap: 7.59e-01 Pobj: -2.4490536e-01 Ad: 1.00e+00 Dobj: -2.4448683e-01 Iter: 39 Ap: 5.16e-01 Pobj: -2.4490287e-01 Ad: 1.00e+00 Dobj: -2.4468839e-01 Iter: 40 Ap: 6.11e-01 Pobj: -2.4490086e-01 Ad: 1.00e+00 Dobj: -2.4477146e-01 Iter: 41 Ap: 7.47e-01 Pobj: -2.4489952e-01 Ad: 1.00e+00 Dobj: -2.4481525e-01 Iter: 42 Ap: 8.37e-01 Pobj: -2.4489870e-01 Ad: 1.00e+00 Dobj: -2.4486045e-01 Iter: 43 Ap: 4.63e-01 Pobj: -2.4489854e-01 Ad: 6.24e-01 Dobj: -2.4486993e-01 Iter: 44 Ap: 7.90e-01 Pobj: -2.4489831e-01 Ad: 1.00e+00 Dobj: -2.4488374e-01 Iter: 45 Ap: 7.92e-01 Pobj: -2.4489819e-01 Ad: 1.00e+00 Dobj: -2.4488939e-01 Iter: 46 Ap: 8.07e-01 Pobj: -2.4489811e-01 Ad: 1.00e+00 Dobj: -2.4489335e-01 Iter: 47 Ap: 8.94e-01 Pobj: -2.4489806e-01 Ad: 1.00e+00 Dobj: -2.4489536e-01 Iter: 48 Ap: 1.00e+00 Pobj: -2.4489800e-01 Ad: 1.00e+00 Dobj: -2.4489734e-01 Iter: 49 Ap: 1.00e+00 Pobj: -2.4489797e-01 Ad: 1.00e+00 Dobj: -2.4489786e-01 Iter: 50 Ap: 1.00e+00 Pobj: -2.4489796e-01 Ad: 9.61e-01 Dobj: -2.4489794e-01 Iter: 51 Ap: 9.56e-01 Pobj: -2.4489796e-01 Ad: 9.37e-01 Dobj: -2.4489796e-01 Success: SDP solved Primal objective value: -2.4489796e-01 Dual objective value: -2.4489796e-01 Relative primal infeasibility: 5.02e-12 Relative dual infeasibility: 9.45e-10 Real Relative Gap: -2.53e-10 XZ Relative Gap: 6.85e-09 DIMACS error measures: 5.88e-11 0.00e+00 1.79e-09 0.00e+00 -2.53e-10 6.85e-09 Elements time: 1439.978704 Factor time: 36.562572 Other time: 28.530312 Total time: 1505.071587 ============================================================================ Return code is 0 Approximate floating-point bound is 0.24489796 $ sage -python scripts/find_sharp_graphs.py --dir output/k4-f32c5 Floating point bound is 0.244897959230519352. 23 members of H are sharp. 0.244897959215229277 : graph 1 (7:) 0.244897958895631512 : graph 2 (7:123) 0.244897959182714009 : graph 3 (7:123124) 0.244897959182621833 : graph 6 (7:123124125) 0.244897959125184250 : graph 15 (7:123124125126) 0.244897959201604620 : graph 20 (7:123124135145) 0.244897959204033039 : graph 46 (7:123124125126127) 0.244897959211505478 : graph 151 (7:123124125136146156) 0.244897959227894868 : graph 1041 (7:123124125126137147157167) 0.244897959211676036 : graph 1042 (7:123124125126137147157267) 0.244897959201865106 : graph 1044 (7:123124125126137147257267) 0.244897959182976993 : graph 1055 (7:123124125126137347357367) 0.244897959126038650 : graph 1676 (7:123124135145236246356456) 0.244897959215973654 : graph 2092 (7:123124125136137146147156157) 0.244897959221518330 : graph 2094 (7:123124125136137146147256257) 0.244897959211138494 : graph 3191 (7:123124125126137147157367467567) 0.244897959220840511 : graph 3193 (7:123124125126137147357367457467) 0.244897959221293537 : graph 3195 (7:123124125136137146147156257567) 0.244897959220782668 : graph 3847 (7:123124125136137146147356357456457) 0.244897959221235362 : graph 4023 (7:123124125136146156237247257367467567) 0.244897959202078963 : graph 4025 (7:123124125136146156237247357367457467) 0.244897959230519352 : graph 4027 (7:123124125136146157237247267356456567) 0.244897959037931962 : graph 4045 (7:123124135146157167236247256257345347367456) Written sharp graphs to flags.py $ sage -python scripts/check_construction.py --n 7 --r 3 --vertex-transitive 7:123124135146157167236247256257345347367456 Density is 12/49. 23 graphs of order 7 occur as induced subgraphs of the blow-up: 7: has density 5797/117649 (0.049274) 7:123 has density 60/16807 (0.003570) 7:123124 has density 360/16807 (0.021420) 7:123124125 has density 360/16807 (0.021420) 7:123124135145 has density 540/16807 (0.032129) 7:123124125126 has density 180/16807 (0.010710) 7:123124125126127 has density 576/16807 (0.034271) 7:123124125136146156 has density 720/16807 (0.042839) 7:123124125126137147257267 has density 540/16807 (0.032129) 7:123124135145236246356456 has density 180/16807 (0.010710) 7:123124125126137347357367 has density 360/16807 (0.021420) 7:123124125126137147157167 has density 1620/16807 (0.096388) 7:123124125126137147157267 has density 720/16807 (0.042839) 7:123124125136137146147156157 has density 120/2401 (0.049979) 7:123124125136137146147256257 has density 1080/16807 (0.064259) 7:123124125136137146147156257567 has density 1080/16807 (0.064259) 7:123124125126137147357367457467 has density 1080/16807 (0.064259) 7:123124125126137147157367467567 has density 720/16807 (0.042839) 7:123124125136137146147356357456457 has density 1080/16807 (0.064259) 7:123124125136146156237247257367467567 has density 1080/16807 (0.064259) 7:123124125136146156237247357367457467 has density 540/16807 (0.032129) 7:123124125136146157237247267356456567 has density 2160/16807 (0.128518) 7:123124135146157167236247256257345347367456 has density 720/117649 (0.006120) $ sage -python scripts/make_zero_eigenvectors.py --vertex-transitive 7:123124135146157167236247256257345347367456 --dir output/k4-f32c5 Constructed 1 out of 1 zero eigenvectors for type 1. Constructed 7 out of 7 zero eigenvectors for type 2. Constructed 2 out of 2 zero eigenvectors for type 3. Constructed 7 out of 7 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 5 out of 5 zero eigenvectors for type 7. Constructed 1 out of 1 zero eigenvectors for type 8. Written zev.py Written field to flags.py $ sage -python scripts/factor_approximate_q.py --dir output/k4-f32c5 Floating point bound is 0.244897959230519296. Type 1: smallest eigenvalue is 0.001138204428361353 Type 2: smallest eigenvalue is 0.011938346823730765 Type 3: smallest eigenvalue is 0.025160683784319304 Type 4: smallest eigenvalue is 0.067026689331544237 Type 5: smallest eigenvalue is 0.059755453711230866 Type 6: smallest eigenvalue is 0.060018333031833182 Type 7: smallest eigenvalue is 0.063597428550280111 Type 8: smallest eigenvalue is 0.060769199496128808 Written r.py Written qdashf.py $ sage -python scripts/make_exact_qdash.py '12/49' --denominator 720 --dir output/k4-f32c5 --diagonalize Type 1: smallest eigenvalue is 0.001235942445228616 Type 2: smallest eigenvalue is 0.005411008939460605 Type 3: smallest eigenvalue is 0.026144532438152249 Type 4: smallest eigenvalue is 0.058277689099283110 Type 5: smallest eigenvalue is 0.059163869850181044 Type 6: smallest eigenvalue is 0.061597680534618456 Type 7: smallest eigenvalue is 0.026103780748441403 Type 8: smallest eigenvalue is 0.061488752732437292 Diagonalizing matrices... Written qdash.py Written r.py Added exact bound to flags.py $ sage -python scripts/verify_bound.py --dir output/k4-f32c5 [True, True, True, True, True, True, True, True] Written q.py Floating point bound (non-sharp graphs) is 0.244584432487519932 Exact bound (just sharp graphs) is 12/49 Bound (all graphs) is 12/49 $ sage -python scripts/make_certificate.py --dir output/k4-f32c5 Written certificate to cert.js