== 1. complete graphs K_n: max abs deviation of numeric tau from (n-1)(1+(n-2)p)/(n-p), and spectrum check
n=  2  max|tau-formula|=0.00e+00  max|spectrum,mu - predicted|=7.22e-16
n=  3  max|tau-formula|=1.55e-15  max|spectrum,mu - predicted|=1.55e-15
n=  4  max|tau-formula|=3.11e-15  max|spectrum,mu - predicted|=2.58e-15
n=  5  max|tau-formula|=4.44e-15  max|spectrum,mu - predicted|=6.66e-16
n=  6  max|tau-formula|=1.07e-14  max|spectrum,mu - predicted|=5.55e-16
n=  8  max|tau-formula|=1.51e-14  max|spectrum,mu - predicted|=7.77e-16
n= 10  max|tau-formula|=4.09e-14  max|spectrum,mu - predicted|=1.14e-15
n= 16  max|tau-formula|=1.55e-13  max|spectrum,mu - predicted|=6.66e-16
n= 32  max|tau-formula|=3.77e-13  max|spectrum,mu - predicted|=7.49e-16
n= 64  max|tau-formula|=2.79e-12  max|spectrum,mu - predicted|=1.91e-15

== 2. endpoints: tau(0) (random walk), tau(p) at p = 1-1e-3, 1-1e-5, 1-1e-7, degree d; claim: tau(1-) = d
graph                            n   d     tau(0)  tau(.999)  tau(1-1e-5)  tau(1-1e-7)    endpoint test
cycle C_3                        3   2    0.66667    1.99800    1.9999800    1.9999998       tau(0) < d
cycle C_4                        4   2    1.00000    2.00000    2.0000000    2.0000000       tau(0) < d
cycle C_5                        5   2    1.44721    2.04419    2.0044671    2.0004472       tau(0) < d
cycle C_6                        6   2    2.00000    2.04522    2.0044771    2.0004473       tau(0) = d
cycle C_7                        7   2    2.65597    2.06399    2.0063320    2.0006325       tau(0) > d
cycle C_8                        8   2    3.41421    2.06451    2.0063371    2.0006326       tau(0) > d
cycle C_9                        9   2    4.27432    2.07386    2.0072509    2.0007238       tau(0) > d
cycle C_12                      12   2    7.46410    2.07959    2.0077668    2.0007748       tau(0) > d
cycle C_20                      20   2   20.43173    2.08777    2.0085330    2.0008509       tau(0) > d
cycle C_40                      40   2   81.22382    2.09133    2.0088633    2.0008837       tau(0) > d
hypercube Q_1                    2   1    0.50000    0.99900    0.9999900    0.9999999       tau(0) < d
hypercube Q_2                    4   2    1.00000    2.00000    2.0000000    2.0000000       tau(0) < d
hypercube Q_3                    8   3    1.50000    3.00100    3.0000100    3.0000001       tau(0) < d
hypercube Q_4                   16   4    2.00000    4.00200    4.0000200    4.0000002       tau(0) < d
hypercube Q_5                   32   5    2.50000    5.00300    5.0000300    5.0000003       tau(0) < d
hypercube Q_6                   64   6    3.00000    6.00400    6.0000400    6.0000004       tau(0) < d
hypercube Q_7                  128   7    3.50000    7.00500    7.0000500    7.0000005       tau(0) < d
hypercube Q_8                  256   8    4.00000    8.00600    8.0000600    8.0000006       tau(0) < d
torus Z_5^2                     25   4    2.89443    4.06380    4.0063296    4.0006325       tau(0) < d
torus Z_8^2                     64   4    6.82843    4.09127    4.0089618    4.0008946       tau(0) > d
torus Z_4^3                     64   6    3.00000    6.00400    6.0000400    6.0000004       tau(0) < d
torus Z_5^3                    125   6    4.34164    6.07908    6.0077611    6.0007747       tau(0) < d
circulant Z_15 {1,2}            15   4    4.79244    4.00026    4.0000025    4.0000000       tau(0) > d
circulant Z_21 {1,4,5}          21   6    1.86991    6.00183    6.0000183    6.0000002       tau(0) < d
circulant Z_30 {1,7}            30   4    4.79244    4.08824    4.0089318    4.0008943       tau(0) > d
Z_12 {1,6}                      12   3    3.00000    3.05547    3.0054839    3.0005478       tau(0) = d
S_3 transpositions               6   3    1.00000    3.00000    3.0000000    3.0000000       tau(0) < d
S_4 transpositions              24   6    1.50000    6.00200    6.0000200    6.0000002       tau(0) < d
S_5 transpositions             120  10    2.00000   10.00500   10.0000500   10.0000005       tau(0) < d
S_4 adjacent transp.            24   3    5.12132    3.07941    3.0077649    3.0007748       tau(0) > d
S_5 adjacent transp.           120   4   10.47214    4.10509    4.0102600    4.0010236       tau(0) > d
S_5 star transp.               120   4    4.00000    4.15161    4.0148031    4.0014768       tau(0) = d
dihedral D_7 {r,r^-1,s}         14   3    3.98396    3.07823    3.0077535    3.0007747       tau(0) > d
dihedral D_12 {r,r^-1,s}        24   3   11.19615    3.09701    3.0095077    3.0009489       tau(0) > d
SL_2(Z_3) deg 4                 24   4    3.15470    4.09105    4.0089602    4.0008946       tau(0) < d
SL_2(Z_5) deg 4                120   4    5.23607    4.14657    4.0142958    4.0014260       tau(0) > d
SL_2(Z_7) deg 4                336   4    6.82843    4.16445    4.0160196    4.0015978       tau(0) > d
complete K_4                     4   3    0.75000    2.99700    2.9999700    2.9999997       tau(0) < d
complete K_7                     7   6    0.85714    5.99400    5.9999400    5.9999994       tau(0) < d
complete K_12                   12  11    0.91667   10.98900   10.9998900   10.9999989       tau(0) < d

== 3. shape of tau on the grid (0, 1e-4..0.1 log-spaced, 0.1..0.99, .995, .999, .9999): monotonicity, sup, argmax
graph                            n   d    tau(0)    sup tau  argmax p sup/tau(0) sup/max(tau0,d)  shape
cycle C_3                        3   2    0.6667     1.9998    0.9999     2.9997          0.9999  increasing
cycle C_4                        4   2    1.0000     2.0000    0.9999     2.0000          1.0000  increasing
cycle C_5                        5   2    1.4472     2.4499    0.7300     1.6928          1.2249  NON-MONOTONE
cycle C_6                        6   2    2.0000     2.9469    0.4800     1.4734          1.4734  NON-MONOTONE
cycle C_7                        7   2    2.6560     3.6530    0.3500     1.3754          1.3754  NON-MONOTONE
cycle C_8                        8   2    3.4142     4.4582    0.2400     1.3058          1.3058  NON-MONOTONE
cycle C_9                        9   2    4.2743     5.3927    0.1800     1.2617          1.2617  NON-MONOTONE
cycle C_12                      12   2    7.4641     8.8490    0.0794     1.1855          1.1855  NON-MONOTONE
cycle C_20                      20   2   20.4317    22.8126    0.0200     1.1165          1.1165  NON-MONOTONE
cycle C_40                      40   2   81.2238    87.5755    0.0040     1.0782          1.0782  NON-MONOTONE
hypercube Q_1                    2   1    0.5000     0.9999    0.9999     1.9998          0.9999  increasing
hypercube Q_2                    4   2    1.0000     2.0000    0.9999     2.0000          1.0000  increasing
hypercube Q_3                    8   3    1.5000     3.1145    0.7700     2.0763          1.0382  NON-MONOTONE
hypercube Q_4                   16   4    2.0000     4.4422    0.5900     2.2211          1.1106  NON-MONOTONE
hypercube Q_5                   32   5    2.5000     5.9835    0.4700     2.3934          1.1967  NON-MONOTONE
hypercube Q_6                   64   6    3.0000     7.7479    0.3800     2.5826          1.2913  NON-MONOTONE
hypercube Q_7                  128   7    3.5000     9.7468    0.3100     2.7848          1.3924  NON-MONOTONE
hypercube Q_8                  256   8    4.0000    11.9928    0.2600     2.9982          1.4991  NON-MONOTONE
torus Z_5^2                     25   4    2.8944     5.7929    0.4200     2.0014          1.4482  NON-MONOTONE
torus Z_8^2                     64   4    6.8284    10.9537    0.1400     1.6041          1.6041  NON-MONOTONE
torus Z_4^3                     64   6    3.0000     7.7479    0.3800     2.5826          1.2913  NON-MONOTONE
torus Z_5^3                    125   6    4.3416    10.3152    0.2600     2.3759          1.7192  NON-MONOTONE
circulant Z_15 {1,2}            15   4    4.7924     6.2182    0.1800     1.2975          1.2975  NON-MONOTONE
circulant Z_21 {1,4,5}          21   6    1.8699     6.1864    0.7900     3.3084          1.0311  NON-MONOTONE
circulant Z_30 {1,7}            30   4    4.7924     6.8911    0.2600     1.4379          1.4379  NON-MONOTONE
Z_12 {1,6}                      12   3    3.0000     4.5371    0.3900     1.5124          1.5124  NON-MONOTONE
S_3 transpositions               6   3    1.0000     3.0000    0.9999     3.0000          1.0000  increasing
S_4 transpositions              24   6    1.5000     6.2665    0.7400     4.1777          1.0444  NON-MONOTONE
S_5 transpositions             120  10    2.0000    11.4806    0.5200     5.7403          1.1481  NON-MONOTONE
S_4 adjacent transp.            24   3    5.1213     7.5824    0.1800     1.4805          1.4805  NON-MONOTONE
S_5 adjacent transp.           120   4   10.4721    16.1853    0.0794     1.5456          1.5456  NON-MONOTONE
S_5 star transp.               120   4    4.0000    11.7563    0.2300     2.9391          2.9391  NON-MONOTONE
dihedral D_7 {r,r^-1,s}         14   3    3.9840     5.5482    0.2700     1.3926          1.3926  NON-MONOTONE
dihedral D_12 {r,r^-1,s}        24   3   11.1962    13.2723    0.0631     1.1854          1.1854  NON-MONOTONE
SL_2(Z_3) deg 4                 24   4    3.1547     6.5984    0.3900     2.0916          1.6496  NON-MONOTONE
SL_2(Z_5) deg 4                120   4    5.2361    12.0431    0.2100     2.3000          2.3000  NON-MONOTONE
SL_2(Z_7) deg 4                336   4    6.8284    18.0136    0.1300     2.6380          2.6380  NON-MONOTONE
complete K_4                     4   3    0.7500     2.9997    0.9999     3.9996          0.9999  increasing
complete K_7                     7   6    0.8571     5.9994    0.9999     6.9993          0.9999  increasing
complete K_12                   12  11    0.9167    10.9989    0.9999    11.9988          0.9999  increasing

== 4. lower bounds: min over grid (p>0) of tau/L1 and tau/L2 (both must be >= 1), and tau/(2 Var dist)
cycle C_3                    min tau/L1=  1.00000  min tau/L2=  1.00000  min tau/(2Var)=  1.50010  mu(e)=max: True
cycle C_4                    min tau/L1=  1.00005  min tau/L2=  1.00000  min tau/(2Var)=  1.00010  mu(e)=max: True
cycle C_5                    min tau/L1=  1.00710  min tau/L2=  1.00700  min tau/(2Var)=  1.29227  mu(e)=max: True
cycle C_6                    min tau/L1=  1.00715  min tau/L2=  1.00705  min tau/(2Var)=  1.09108  mu(e)=max: True
cycle C_7                    min tau/L1=  1.01009  min tau/L2=  1.00999  min tau/(2Var)=  1.25155  mu(e)=max: True
cycle C_8                    min tau/L1=  1.01011  min tau/L2=  1.01001  min tau/(2Var)=  1.13832  mu(e)=max: True
cycle C_9                    min tau/L1=  1.01157  min tau/L2=  1.01146  min tau/(2Var)=  1.23677  mu(e)=max: True
cycle C_12                   min tau/L1=  1.01240  min tau/L2=  1.01230  min tau/(2Var)=  1.17900  mu(e)=max: True
cycle C_20                   min tau/L1=  1.01363  min tau/L2=  1.01353  min tau/(2Var)=  1.20283  mu(e)=max: True
cycle C_40                   min tau/L1=  1.01417  min tau/L2=  1.01406  min tau/(2Var)=  1.21514  mu(e)=max: True
hypercube Q_1                min tau/L1=  1.00000  min tau/L2=  1.00000  min tau/(2Var)=  1.00005  mu(e)=max: True
hypercube Q_2                min tau/L1=  1.00005  min tau/L2=  1.00000  min tau/(2Var)=  1.00010  mu(e)=max: True
hypercube Q_3                min tau/L1=  1.00007  min tau/L2=  1.00000  min tau/(2Var)=  1.00015  mu(e)=max: True
hypercube Q_4                min tau/L1=  1.00008  min tau/L2=  1.00000  min tau/(2Var)=  1.00020  mu(e)=max: True
hypercube Q_5                min tau/L1=  1.00008  min tau/L2=  1.00000  min tau/(2Var)=  1.00025  mu(e)=max: True
hypercube Q_6                min tau/L1=  1.00008  min tau/L2=  1.00000  min tau/(2Var)=  1.00030  mu(e)=max: True
hypercube Q_7                min tau/L1=  1.00009  min tau/L2=  1.00000  min tau/(2Var)=  1.00035  mu(e)=max: True
hypercube Q_8                min tau/L1=  1.00009  min tau/L2=  1.00000  min tau/(2Var)=  1.00041  mu(e)=max: True
torus Z_5^2                  min tau/L1=  1.00504  min tau/L2=  1.00494  min tau/(2Var)=  1.29232  mu(e)=max: True
torus Z_8^2                  min tau/L1=  1.00714  min tau/L2=  1.00704  min tau/(2Var)=  1.13852  mu(e)=max: True
torus Z_4^3                  min tau/L1=  1.00008  min tau/L2=  1.00000  min tau/(2Var)=  1.00030  mu(e)=max: True
torus Z_5^3                  min tau/L1=  1.00412  min tau/L2=  1.00402  min tau/(2Var)=  1.29233  mu(e)=max: True
circulant Z_15 {1,2}         min tau/L1=  1.00007  min tau/L2=  1.00002  min tau/(2Var)=  1.82190  mu(e)=max: True
circulant Z_21 {1,4,5}       min tau/L1=  1.00008  min tau/L2=  1.00001  min tau/(2Var)=  1.90930  mu(e)=max: True
circulant Z_30 {1,7}         min tau/L1=  1.00707  min tau/L2=  1.00696  min tau/(2Var)=  1.62038  mu(e)=max: True
Z_12 {1,6}                   min tau/L1=  1.00583  min tau/L2=  1.00574  min tau/(2Var)=  1.64888  mu(e)=max: True
S_3 transpositions           min tau/L1=  1.00003  min tau/L2=  1.00003  min tau/(2Var)=  1.05903  mu(e)=max: True
S_4 transpositions           min tau/L1=  1.00005  min tau/L2=  1.00003  min tau/(2Var)=  1.13740  mu(e)=max: True
S_5 transpositions           min tau/L1=  1.00006  min tau/L2=  1.00003  min tau/(2Var)=  1.22117  mu(e)=max: True
S_4 adjacent transp.         min tau/L1=  1.00826  min tau/L2=  1.00815  min tau/(2Var)=  1.18221  mu(e)=max: True
S_5 adjacent transp.         min tau/L1=  1.00818  min tau/L2=  1.00807  min tau/(2Var)=  1.25727  mu(e)=max: True
S_5 star transp.             min tau/L1=  1.01180  min tau/L2=  1.01164  min tau/(2Var)=  1.53977  mu(e)=max: True
dihedral D_7 {r,r^-1,s}      min tau/L1=  1.00822  min tau/L2=  1.00813  min tau/(2Var)=  1.51938  mu(e)=max: True
dihedral D_12 {r,r^-1,s}     min tau/L1=  1.01010  min tau/L2=  1.01001  min tau/(2Var)=  1.63928  mu(e)=max: True
SL_2(Z_3) deg 4              min tau/L1=  1.00716  min tau/L2=  1.00706  min tau/(2Var)=  1.84664  mu(e)=max: True
SL_2(Z_5) deg 4              min tau/L1=  1.01140  min tau/L2=  1.01124  min tau/(2Var)=  1.90312  mu(e)=max: True
SL_2(Z_7) deg 4              min tau/L1=  1.01277  min tau/L2=  1.01262  min tau/(2Var)=  2.16670  mu(e)=max: True
complete K_4                 min tau/L1=  1.00000  min tau/L2=  1.00000  min tau/(2Var)=  2.00015  mu(e)=max: True
complete K_7                 min tau/L1=  1.00000  min tau/L2=  1.00000  min tau/(2Var)=  3.50030  mu(e)=max: True
complete K_12                min tau/L1=  1.00000  min tau/L2=  1.00000  min tau/(2Var)=  6.00055  mu(e)=max: True
