association-schemes package

This is a Sagemath package for association schemes. To load the package, just type the following in a terminal.

load("https://raw.githubusercontent.com/sarobidy19/association-schemes/refs/heads/main/code.sage")

Generic association schemes

An association scheme in the association-schemes package is defined by giving the adjacency matrices corresponding to the relations.

association_scheme(adjacency_matrices)
INPUT:
 - adjacency_matrices: a list of 01-matrices forming an association scheme.
OUTPUT:
  A class called association_scheme.

Example: The Johnson scheme J(5,2) can be obtained as follows.

sage: X = graphs.PetersenGraph()
sage: A = X.adjacency_matrix()
sage: B = X.complement().adjacency_matrix()
sage: I = matrix.identity(X.order())
sage: AS = association_scheme([I,A,B])

Methods

• order()

  Return the number of vertices in self.

    sage: X = graphs.PetersenGraph()
    sage: A = X.adjacency_matrix()
    sage: B = X.complement().adjacency_matrix()
    sage: I = matrix.identity(X.order())
    sage: AS = association_scheme([I,A,B])
    sage: AS.order()
    10
  
  

• rank()

  Return the number of relations in self.

    sage: X = graphs.PetersenGraph()
    sage: A = X.adjacency_matrix()
    sage: B = X.complement().adjacency_matrix()
    sage: I = matrix.identity(X.order())
    sage: AS = association_scheme([I,A,B])
    sage: AS.rank()
    3
  

• adjacency_matrices()

  Return the adjacency matrices of self as a list.

    sage: AS.adjacency_matrices()
    [
    [1 0 0 0 0 0 0 0 0 0]  [0 1 0 0 1 1 0 0 0 0]  [0 0 1 1 0 0 1 1 1 1]
    [0 1 0 0 0 0 0 0 0 0]  [1 0 1 0 0 0 1 0 0 0]  [0 0 0 1 1 1 0 1 1 1]
    [0 0 1 0 0 0 0 0 0 0]  [0 1 0 1 0 0 0 1 0 0]  [1 0 0 0 1 1 1 0 1 1]
    [0 0 0 1 0 0 0 0 0 0]  [0 0 1 0 1 0 0 0 1 0]  [1 1 0 0 0 1 1 1 0 1]
    [0 0 0 0 1 0 0 0 0 0]  [1 0 0 1 0 0 0 0 0 1]  [0 1 1 0 0 1 1 1 1 0]
    [0 0 0 0 0 1 0 0 0 0]  [1 0 0 0 0 0 0 1 1 0]  [0 1 1 1 1 0 1 0 0 1]
    [0 0 0 0 0 0 1 0 0 0]  [0 1 0 0 0 0 0 0 1 1]  [1 0 1 1 1 1 0 1 0 0]
    [0 0 0 0 0 0 0 1 0 0]  [0 0 1 0 0 1 0 0 0 1]  [1 1 0 1 1 0 1 0 1 0]
    [0 0 0 0 0 0 0 0 1 0]  [0 0 0 1 0 1 1 0 0 0]  [1 1 1 0 1 0 0 1 0 1]
    [0 0 0 0 0 0 0 0 0 1], [0 0 0 0 1 0 1 1 0 0], [1 1 1 1 0 1 0 0 1 0]
    ]
  

• base_matrix()

  Return the base matrix of self. If \((\Omega,\mathcal{R})\) is an association scheme with adjacency matrices \(A_0 = I, A_1,\ldots, A_d\), then the base matrix of \((\Omega,\mathcal{R})\) is the matrix \(0A_0 + 1A_1+2A_2+ \ldots+ dA_d\).

    sage: X = graphs.PetersenGraph()
    sage: A = X.adjacency_matrix()
    sage: B = X.complement().adjacency_matrix()
    sage: I = matrix.identity(X.order())
    sage: AS = association_scheme([I,A,B])
    sage: AS.base_matrix()
    [0 1 2 2 1 1 2 2 2 2]
    [1 0 1 2 2 2 1 2 2 2]
    [2 1 0 1 2 2 2 1 2 2]
    [2 2 1 0 1 2 2 2 1 2]
    [1 2 2 1 0 2 2 2 2 1]
    [1 2 2 2 2 0 2 1 1 2]
    [2 1 2 2 2 2 0 2 1 1]
    [2 2 1 2 2 1 2 0 2 1]
    [2 2 2 1 2 1 1 2 0 2]
    [2 2 2 2 1 2 1 1 2 0]
  

• intersection_number(i,j,k)

  Return the intersection number \(p_{ij}^k\) of the association scheme self.

  INPUT: integers \(i,j,\) and \(k\) between \(0\) and the \(r\), where \(r+1\) is the rank of the association scheme.

  OUTPUT: the value of \(p_{ij}^k\).

  EXAMPLE:

  For example, the intersection numbers of the affine polar graph \(VO_6^-(2)\) can be computed as follows.

        sage: X = graphs.AffineOrthogonalPolarGraph(6,2,sign="-")
        sage: A = X.adjacency_matrix()
        sage: B = X.complement().adjacency_matrix()
        sage: I = matrix.identity(X.order())
        sage: AS = association_scheme([I,A,B])
        sage: AS.intersection_number(0,1,1)
        1
        sage: AS.intersection_number(1,1,1)
        10
        sage: AS.intersection_number(2,2,1)
        20
        sage: AS.intersection_number(2,2,2)
        20
  

• is_commutative()

  Return whether or not self is a commutative association scheme.

  The \(d\)-class assocition scheme \((\Omega,\mathcal{R})\) is commutative if its intersection numbers satisfy \(p_{ij}^k = p_{ji}^k\), for all \(0\leq i,j,k\leq d\).

  EXAMPLE:

    sage: AS = OrbitalSchemeTransitiveGroup(group_acting_on_subsets(PSL(2,7),2))
    sage: AS.is_commutative()
    False
    sage: AS = OrbitalSchemeTransitiveGroup(group_acting_on_subsets(AlternatingGroup(7),2))
    sage: AS.is_commutative()
    True
  
  

• is_schurian()

  Return whether or not self is Schurian, that is, its relations are the orbital of a transitive group

  EXAMPLE:

    sage: X = graphs.ShrikhandeGraph()
    sage: G = X.automorphism_group()
    sage: AS1 = OrbitalSchemeTransitiveGroup(G)
    sage: AS1.is_schurian()
    True
    sage: A = X.adjacency_matrix()
    sage: B = X.complement().adjacency_matrix()
    sage: I = matrix.identity(X.order())
    sage: AS2 = association_scheme([I,A,B])
    sage: AS2.is_schurian()
    False
  

• automorphism_group()

  Return the automorphism group of self, that is, the permutation group that preserves all relations of self.

  EXAMPLE:

    sage: X = graphs.ShrikhandeGraph()
    sage: G = X.automorphism_group()
    sage: A = X.adjacency_matrix()
    sage: B = X.complement().adjacency_matrix()
    sage: I = matrix.identity(X.order())
    sage: AS = association_scheme([I,A,B])
    sage: K = AS.automorphism_group()
    sage: K.structure_description()
    '(((C4 x C4) : C3) : C2) : C2'
    sage: K.is_transitive()
    True
  

• character_table()
• P_matrix()
• Q_matrix()
• is_formally_self_dual()
• dimension_of_t_zero()
• dimension_of_centralizer_algebra()
• TerwilligerAlgebra()
• graphs_in_scheme()
• ratio_bound()