You may use these examples in the SAGE codes on the right by setting the id variable accordingly.
| d | r | nr-1...n1 | bound | id |
|---|---|---|---|---|
| 3 | 2 | 7 | 0.450789935825930 | 0 |
| 3 | 2 | 11 | 0.450789935825245 | 1 |
| 3 | 3 | 5x4 | 0.450786019758619 | 2 |
| 3 | 3 | 7x3 | 0.450786019542188 | 3 |
| 3 | 3 | 6x4 | 0.450786017530902 | 4 |
| 3 | 3 | 8x3 | 0.450786017332501 | 5 |
| 3 | 3 | 8x4 | 0.450786017199332 | 6 |
| 3 | 4 | 6x2x2 | 0.450785362258564 | 7 |
| 3 | 4 | 4x3x2 | 0.450785361412891 | 8 |
| 3 | 4 | 8x2x2 | 0.450785345462406 | 9 |
| 3 | 5 | 3x2x2x2 | 0.450785341585753 | 10 |
| 3 | 5 | 4x2x2x2 | 0.450785209433452 | 11 |
| 4 | 2 | 5 | 0.411100755214762 | 12 |
| 4 | 2 | 7 | 0.411100755108140 | 13 |
| 4 | 2 | 10 | 0.411100755012991 | 14 |
| 4 | 2 | 18 | 0.411100754978387 | 15 |
| 4 | 3 | 6x2 | 0.411095104158197 | 16 |
| 4 | 3 | 6x4 | 0.411095100604367 | 17 |
| 4 | 4 | 4x2x2 | 0.411094299550269 | 18 |
| 4 | 4 | 5x2x2 | 0.411094157063507 | 19 |
| 4 | 4 | 4x3x2 | 0.411094130387406 | 20 |
| 5 | 2 | 6 | 0.379176249255960 | 21 |
| 5 | 2 | 8 | 0.379176249238348 | 22 |
| 5 | 3 | 3x3 | 0.379170371133127 | 23 |
| 5 | 4 | 2x2x3 | 0.379170309146112 | 24 |
| 6 | 2 | 5 | 0.352905513275779 | 25 |
| 6 | 2 | 7 | 0.352905513035201 | 26 |
| 6 | 3 | 4x2 | 0.352900231215215 | 27 |
| 6 | 4 | 2x2x2 | 0.352900050015635 | 28 |
| 6 | 4 | 3x2x2 | 0.352899484556593 | 29 |
| 7 | 2 | 5 | 0.330821476921557 | 30 |
| 7 | 3 | 3x2 | 0.330817320344994 | 31 |
| 7 | 3 | 4x2 | 0.330817057782614 | 32 |
| 7 | 3 | 5x2 | 0.330817013868137 | 33 |
| 8 | 2 | 4 | 0.311925393564340 | 34 |
| 8 | 2 | 5 | 0.311925393481482 | 35 |
| 8 | 2 | 6 | 0.311925386189954 | 36 |
| 8 | 3 | 3x2 | 0.311922226926807 | 37 |
| 9 | 2 | 4 | 0.295520279183914 | 38 |
| 9 | 2 | 5 | 0.295520272749388 | 39 |
| 9 | 3 | 2x2 | 0.295519496373816 | 40 |
| 10 | 2 | 5 | 0.281105185061509 | 41 |
| 10 | 3 | 2x2 | 0.281104952946450 | 42 |
| 11 | 2 | 5 | 0.268310125607679 | 43 |
| 11 | 2 | 6 | 0.268310123877474 | 44 |
| 11 | 2 | 7 | 0.268310123579359 | 45 |
| 12 | 2 | 5 | 0.256855204987147 | 46 |
| 13 | 2 | 4 | 0.246524236058568 | 47 |
| 13 | 2 | 6 | 0.246524235446418 | 48 |
| 14 | 2 | 4 | 0.237147192118915 | 49 |
| 15 | 2 | 4 | 0.228587913167835 | 50 |
| 16 | 2 | 4 | 0.220736277869352 | 51 |
This page is for the paper Improved replica bounds for the independence ratio of random regular graphs, where I tried to find r-step replica symmetry breaking (r-RSB) upper bounds for the asymptotic independence ratio of random d-regular graphs.
Follow the links on the left for the formulas and for program codes. Or choose an exmaple from the table on the left and try some of the SAGE codes below by setting the id variable accordingly.
With check.sage one can compute an r-RSB bound for a given set of parameters to check the claimed bounds. You can set the precision into any number of bits and compute the bound using RealField(bits) numbers.
You can explore the parameters with your own code.
With rsbopt.sage one can try to find new bounds. In a SageMath cell the running time is very limited so the optimization will terminate here only for small values of d, r, nrs.