graph                         n   d    gap(0) dimV   c(zonal)      c_max  slope p=1e-5  slope p=1e-6 min u(x), x!=e
cycle C_3                     3   2  1.500000    2    2.00000    2.00000       1.99998       2.00000       0.666667
cycle C_4                     4   2  1.000000    2    1.50000    1.50000       1.49998       1.50000       0.750000
cycle C_5                     5   2  0.690983    2    1.38197    1.38197       1.38193       1.38196       0.800000
cycle C_6                     6   2  0.500000    2    1.25000    1.25000       1.24995       1.24999       0.833333
cycle C_7                     7   2  0.376510    2    1.16201    1.16201       1.16194       1.16200       0.857143
cycle C_8                     8   2  0.292893    2    1.08579    1.08579       1.08569       1.08578       0.875000
cycle C_9                     9   2  0.233956    2    1.02726    1.02726       1.02714       1.02725       0.888889
cycle C_12                   12   2  0.133975    2    0.90192    0.90192       0.90173       0.90190       0.916667
cycle C_20                   20   2  0.048943    2    0.74472    0.74472       0.74421       0.74467       0.950000
cycle C_40                   40   2  0.012312    2    0.62312    0.62312       0.62126       0.62293       0.975000
hypercube Q_1                 2   1  2.000000    1    1.00000    1.00000       1.00000       1.00000       0.500000
hypercube Q_2                 4   2  1.000000    2    1.50000    1.50000       1.49998       1.50000       0.750000
hypercube Q_3                 8   3  0.666667    3    2.00000    2.00000       1.99995       2.00000       0.875000
hypercube Q_4                16   4  0.500000    4    2.50000    2.50000       2.49994       2.49999       0.937500
hypercube Q_5                32   5  0.400000    5    3.00000    3.00000       2.99997       3.00000       0.968750
hypercube Q_6                64   6  0.333333    6    3.50000    3.50000       3.50015       3.50001       0.984375
hypercube Q_7               128   7  0.285714    7    4.00000    4.00000       4.00059       4.00006       0.992187
hypercube Q_8               256   8  0.250000    8    4.50000    4.50000       4.50154       4.50015       0.996094
torus Z_5^2                  25   4  0.345492    4    2.38197    2.38197       2.38184       2.38195       0.960000
torus Z_8^2                  64   4  0.146447    4    2.08579    2.08579       2.08541       2.08575       0.984375
torus Z_4^3                  64   6  0.333333    6    3.50000    3.50000       3.50015       3.50001       0.984375
torus Z_5^3                 125   6  0.230328    6    3.38197    3.38197       3.38208       3.38198       0.992000
circulant Z_15 {1,2}         15   4  0.208662    2    1.06475    1.06475       1.06462       1.06473       0.867760
circulant Z_21 {1,4,5}       21   6  0.534785    6    4.27687    4.27687       4.27665       4.27684       0.952381
circulant Z_30 {1,7}         30   4  0.208662    2    1.07734    1.07734       1.07725       1.07733       0.956868
Z_12 {1,6}                   12   3  0.333333    2    1.25000    1.25000       1.24992       1.24999       0.865385
S_3 transpositions            6   3  1.000000    4    3.33333    3.33333       3.33323       3.33332       0.833333
S_4 transpositions           24   6  0.666667    9    6.62500    6.62500       6.62503       6.62500       0.958333
S_5 transpositions          120  10  0.500000   16   10.90000   10.90000      10.90530      10.90053       0.991667
S_4 adjacent transp.         24   3  0.195262    3    1.62763    1.62894       1.62871       1.62892       0.930060
S_5 adjacent transp.        120   4  0.095492    4    2.03456    2.03515       2.03455       2.03509       0.964739
S_5 star transp.            120   4  0.250000   12    7.30000    7.30000       7.29952       7.29995       0.991667
dihedral D_7                 14   3  0.251007    2    1.16201    1.16201       1.16190       1.16200       0.866197
dihedral D_12                24   3  0.089316    2    0.90192    0.90192       0.90162       0.90189       0.866026
SL_2(Z_3) deg 4              24   4  0.316987    4    2.48618    2.48618       2.48608       2.48617       0.958333
SL_2(Z_5) deg 4             120   4  0.190983    3    1.66512    1.66512       1.66608       1.66522       0.991667
SL_2(Z_7) deg 4             336   4  0.146447    8    4.63789    4.63822       4.64063       4.63846       0.997024
complete K_2                  2   1  2.000000    1    1.00000    1.00000       1.00000       1.00000       0.500000
complete K_4                  4   3  1.333333    3    3.00000    3.00000       2.99994       2.99999       0.750000
complete K_7                  7   6  1.166667    6    6.00000    6.00000       5.99970       5.99997       0.857143
complete K_12                12  11  1.090909   11   11.00000   11.00000      10.99890      10.99989       0.916667

all graphs: c(zonal) > 0 and numerical slope of the gap negative (tau increasing at 0+): True
