#  July 9, 2008.
#
#  Q-bipartite double of complement Higman-Sims graph: P-polynomial
#  (This is called the "double Higman-Sims" graph in [BCN].)
#
#  SRG has
#      [  1   77  22  ]        [ 1   22     77 ]
#  P = [  1    7  -8  ]   Q =  [ 1    2     -3 ]
#      [  1   -3   2  ]        [ 1   -8      7 ]
d := 5;

v := [1, 22, 77, 77, 22, 1]:
verts := 200;

Q := matrix(6,6,[
1, 22,  77,  77,  22,  1,
1,  8,   7,  -7,  -8, -1,
1,  2,  -3,  -3,   2,  1,
1, -2,  -3,   3,   2, -1,
1, -8,   7,   7,  -8,  1,
1,-22,  77, -77,  22, -1]);


#  This tridiagonal  matrix, L_1-star,  allows us to fill out the cols of Q
#  if we are only given the cosines (normalized column 1 of Q).
L := matrix([
[0  , 22  ,  0  ,  0  ,  0  , 0],
[1  ,  0  , 21  ,  0  ,  0  , 0],
[0  ,  6  ,  0  , 16  ,  0  , 0],
[0  ,  0  , 16  ,  0  ,  6  , 0],
[0  ,  0  ,  0  , 21  ,  0  , 1],
[0  ,  0  ,  0  ,  0  , 22  , 0]]):
 

 
          [0    22     0     0     0    0]
          [                              ]
          [1     0    21     0     0    0]
          [                              ]
          [0     6     0    16     0    0]
  L_1 =   [                              ]
          [0     0    16     0     6    0]
          [                              ]
          [0     0     0    21     0    1]
          [                              ]
          [0     0     0     0    22    0]
 
          [0     0    77     0     0    0]
          [                              ]
          [0    21     0    56     0    0]
          [                              ]
          [1     0    60     0    16    0]
   L_2 =  [                              ]
          [0    16     0    60     0    1]
          [                              ]
          [0     0    56     0    21    0]
          [                              ]
          [0     0     0    77     0    0]
 
          [0     0     0    77     0    0]
          [                              ]
          [0     0    56     0    21    0]
          [                              ]
          [0    16     0    60     0    1]
   L_3 =  [                              ]
          [1     0    60     0    16    0]
          [                              ]
          [0    21     0    56     0    0]
          [                              ]
          [0     0    77     0     0    0]
 
          [0     0     0     0    22    0]
          [                              ]
          [0     0     0    21     0    1]
          [                              ]
          [0     0    16     0     6    0]
   L_4 =  [                              ]
          [0     6     0    16     0    0]
          [                              ]
          [1     0    21     0     0    0]
          [                              ]
          [0    22     0     0     0    0]
 
              [0    0    0    0    0    1]
              [                          ]
              [0    0    0    0    1    0]
              [                          ]
              [0    0    0    1    0    0]
       L_5 =  [                          ]
              [0    0    1    0    0    0]
              [                          ]
              [0    1    0    0    0    0]
              [                          ]
              [1    0    0    0    0    0]
 
 
       [1     22    77     77    22     1]
       [                                 ]
       [1      8     7     -7    -8    -1]
       [                                 ]
       [1      2    -3     -3     2     1]
  P := [                                 ]
       [1     -2    -3      3     2    -1]
       [                                 ]
       [1     -8     7      7    -8     1]
       [                                 ]
       [1    -22    77    -77    22    -1]

       [1     22    77     77    22     1]
       [                                 ]
       [1      8     7     -7    -8    -1]
       [                                 ]
       [1      2    -3     -3     2     1]
 Q :=  [                                 ]
       [1     -2    -3      3     2    -1]
       [                                 ]
       [1     -8     7      7    -8     1]
       [                                 ]
       [1    -22    77    -77    22    -1]


 
 
      [0    22     0     0     0    0]
      [                              ]
      [1     0    21     0     0    0]
      [                              ]
      [0     6     0    16     0    0]
Ls1 = [                              ]
      [0     0    16     0     6    0]
      [                              ]
      [0     0     0    21     0    1]
      [                              ]
      [0     0     0     0    22    0]
 
      [0     0    77     0     0    0]
      [                              ]
      [0    21     0    56     0    0]
      [                              ]
      [1     0    60     0    16    0]
Ls2 = [                              ]
      [0    16     0    60     0    1]
      [                              ]
      [0     0    56     0    21    0]
      [                              ]
      [0     0     0    77     0    0]
 
      [0     0     0    77     0    0]
      [                              ]
      [0     0    56     0    21    0]
      [                              ]
      [0    16     0    60     0    1]
Ls3 = [                              ]
      [1     0    60     0    16    0]
      [                              ]
      [0    21     0    56     0    0]
      [                              ]
      [0     0    77     0     0    0]
 
      [0     0     0     0    22    0]
      [                              ]
      [0     0     0    21     0    1]
      [                              ]
      [0     0    16     0     6    0]
Ls4 = [                              ]
      [0     6     0    16     0    0]
      [                              ]
      [1     0    21     0     0    0]
      [                              ]
      [0    22     0     0     0    0]
 
      [0    0    0    0    0    1]
      [                          ]
      [0    0    0    0    1    0]
      [                          ]
      [0    0    0    1    0    0]
Ls5 = [                          ]
      [0    0    1    0    0    0]
      [                          ]
      [0    1    0    0    0    0]
      [                          ]
      [1    0    0    0    0    0]