venv/lib/python3.12/site-packages/sympy/matrices/tests/test_graph.py

from sympy.combinatorics import Permutation
from sympy.core.symbol import symbols
from sympy.matrices import Matrix
from sympy.matrices.expressions import (
    PermutationMatrix, BlockDiagMatrix, BlockMatrix)


def test_connected_components():
    a, b, c, d, e, f, g, h, i, j, k, l, m = symbols('a:m')

    M = Matrix([
        [a, 0, 0, 0, b, 0, 0, 0, 0, 0, c, 0, 0],
        [0, d, 0, 0, 0, e, 0, 0, 0, 0, 0, f, 0],
        [0, 0, g, 0, 0, 0, h, 0, 0, 0, 0, 0, i],
        [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
        [m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
        [0, m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
        [0, 0, m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
        [j, 0, 0, 0, k, 0, 0, 1, 0, 0, l, 0, 0],
        [0, j, 0, 0, 0, k, 0, 0, 1, 0, 0, l, 0],
        [0, 0, j, 0, 0, 0, k, 0, 0, 1, 0, 0, l],
        [0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1, 0, 0],
        [0, 0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1, 0],
        [0, 0, 0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1]])
    cc = M.connected_components()
    assert cc == [[0, 4, 7, 10], [1, 5, 8, 11], [2, 6, 9, 12], [3]]

    P, B = M.connected_components_decomposition()
    p = Permutation([0, 4, 7, 10, 1, 5, 8, 11, 2, 6, 9, 12, 3])
    assert P == PermutationMatrix(p)

    B0 = Matrix([
        [a, b, 0, c],
        [m, 1, 0, 0],
        [j, k, 1, l],
        [0, d, 0, 1]])
    B1 = Matrix([
        [d, e, 0, f],
        [m, 1, 0, 0],
        [j, k, 1, l],
        [0, d, 0, 1]])
    B2 = Matrix([
        [g, h, 0, i],
        [m, 1, 0, 0],
        [j, k, 1, l],
        [0, d, 0, 1]])
    B3 = Matrix([[1]])
    assert B == BlockDiagMatrix(B0, B1, B2, B3)


def test_strongly_connected_components():
    M = Matrix([
        [11, 14, 10, 0, 15, 0],
        [0, 44, 0, 0, 45, 0],
        [1, 4, 0, 0, 5, 0],
        [0, 0, 0, 22, 0, 23],
        [0, 54, 0, 0, 55, 0],
        [0, 0, 0, 32, 0, 33]])
    scc = M.strongly_connected_components()
    assert scc == [[1, 4], [0, 2], [3, 5]]

    P, B = M.strongly_connected_components_decomposition()
    p = Permutation([1, 4, 0, 2, 3, 5])
    assert P == PermutationMatrix(p)
    assert B == BlockMatrix([
        [
            Matrix([[44, 45], [54, 55]]),
            Matrix.zeros(2, 2),
            Matrix.zeros(2, 2)
        ],
        [
            Matrix([[14, 15], [4, 5]]),
            Matrix([[11, 10], [1, 0]]),
            Matrix.zeros(2, 2)
        ],
        [
            Matrix.zeros(2, 2),
            Matrix.zeros(2, 2),
            Matrix([[22, 23], [32, 33]])
        ]
    ])
    P = P.as_explicit()
    B = B.as_explicit()
    assert P.T * B * P == M

    P, B = M.strongly_connected_components_decomposition(lower=False)
    p = Permutation([3, 5, 0, 2, 1, 4])
    assert P == PermutationMatrix(p)
    assert B == BlockMatrix([
        [
            Matrix([[22, 23], [32, 33]]),
            Matrix.zeros(2, 2),
            Matrix.zeros(2, 2)
        ],
        [
            Matrix.zeros(2, 2),
            Matrix([[11, 10], [1, 0]]),
            Matrix([[14, 15], [4, 5]])
        ],
        [
            Matrix.zeros(2, 2),
            Matrix.zeros(2, 2),
            Matrix([[44, 45], [54, 55]])
        ]
    ])
    P = P.as_explicit()
    B = B.as_explicit()
    assert P.T * B * P == M