#  July 9, 2008.
#
#  Q-bipartite double of Clebsch graph -- 5-cube
#
#
#  SRG has 
#      [  1   10   5  ]        [ 1   5  10 ]
#  P = [  1    2  -3  ]   Q =  [ 1   1  -2 ]
#      [  1   -2   1  ]        [ 1  -3   2 ]
d := 5;

v := [1, 5, 10, 10, 5, 1]:
verts := 32;

Q := matrix(6,6,[
1,  5,  10,  10,  5,  1,
1,  3,   2,  -2, -3, -1,
1,  1,  -2,  -2,  1,  1,
1, -1,  -2,   2,  1, -1,
1, -3,   2,   2, -3,  1,
1, -5,  10, -10,  5, -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 ,    5  ,   0    ,  0    ,   0    ,  0  ],
[1 ,    0  ,   4    ,  0    ,   0    ,  0  ],
[0 ,    2  ,   0    ,  3    ,   0    ,  0  ],
[0 ,    0  ,   3    ,  0    ,   2    ,  0  ],
[0 ,    0  ,   0    ,  4    ,   0    ,  1  ],
[0 ,    0  ,   0    ,  0    ,   5    ,  0  ]
]);


 
       [0    5    0    0    0    0]
       [                          ]
       [1    0    4    0    0    0]
       [                          ]
       [0    2    0    3    0    0]
L_1 =  [                          ]
       [0    0    3    0    2    0]
       [                          ]
       [0    0    0    4    0    1]
       [                          ]
       [0    0    0    0    5    0]
 
       [0    0    10     0    0    0]
       [                            ]
       [0    4     0     6    0    0]
       [                            ]
       [1    0     6     0    3    0]
L_2 =  [                            ]
       [0    3     0     6    0    1]
       [                            ]
       [0    0     6     0    4    0]
       [                            ]
       [0    0     0    10    0    0]
 
       [0    0     0    10    0    0]
       [                            ]
       [0    0     6     0    4    0]
       [                            ]
       [0    3     0     6    0    1]
L_3 =  [                            ]
       [1    0     6     0    3    0]
       [                            ]
       [0    4     0     6    0    0]
       [                            ]
       [0    0    10     0    0    0]
 
       [0    0    0    0    5    0]
       [                          ]
       [0    0    0    4    0    1]
       [                          ]
       [0    0    3    0    2    0]
L_4 =  [                          ]
       [0    2    0    3    0    0]
       [                          ]
       [1    0    4    0    0    0]
       [                          ]
       [0    5    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     5    10     10     5     1]
       [                                ]
       [1     3     2     -2    -3    -1]
       [                                ]
       [1     1    -2     -2     1     1]
  P := [                                ]
       [1    -1    -2      2     1    -1]
       [                                ]
       [1    -3     2      2    -3     1]
       [                                ]
       [1    -5    10    -10     5    -1]
 


       [1     5    10     10     5     1]
       [                                ]
       [1     3     2     -2    -3    -1]
       [                                ]
       [1     1    -2     -2     1     1]
  Q := [                                ]
       [1    -1    -2      2     1    -1]
       [                                ]
       [1    -3     2      2    -3     1]
       [                                ]
       [1    -5    10    -10     5    -1]
 
 
 
       [0    5    0    0    0    0]
       [                          ]
       [1    0    4    0    0    0]
       [                          ]
       [0    2    0    3    0    0]
Ls1 =  [                          ]
       [0    0    3    0    2    0]
       [                          ]
       [0    0    0    4    0    1]
       [                          ]
       [0    0    0    0    5    0]

       [0    0    10     0    0    0]
       [                            ]
       [0    4     0     6    0    0]
       [                            ]
       [1    0     6     0    3    0]
Ls2 =  [                            ]
       [0    3     0     6    0    1]
       [                            ]
       [0    0     6     0    4    0]
       [                            ]
       [0    0     0    10    0    0]
 
       [0    0     0    10    0    0]
       [                            ]
       [0    0     6     0    4    0]
       [                            ]
       [0    3     0     6    0    1]
Ls3 =  [                            ]
       [1    0     6     0    3    0]
       [                            ]
       [0    4     0     6    0    0]
       [                            ]
       [0    0    10     0    0    0]
 
       [0    0    0    0    5    0]
       [                          ]
       [0    0    0    4    0    1]
       [                          ]
       [0    0    3    0    2    0]
Ls4 =  [                          ]
       [0    2    0    3    0    0]
       [                          ]
       [1    0    4    0    0    0]
       [                          ]
       [0    5    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]