Let be a graph. A divisor on is an element of the free group generated by the vertices , .

We say that divisors and are linearly equivalent and write if is a principal divisor, i.e., if for some function . Note that if and are linearly equivalent, they must have the same degree, since the degree of every principal divisor is 0. Divisors of degree 0 are linearly equivalent if and only if they determine the same element of the Jacobian. If is a divisor of degree 0, we denote by the element of the Jacobian determined by . A divisor is said to be effective if for all vertices . We write to mean that is effective. The linear system associated to a divisor is the set

i.e., is the set of all effective divisors linearly equivalent to . Note that if , then . We note also that if , then must be empty.

Sage can be used to compute the linear system of any divisor on a graph.

def linear_system(D, Gamma):
    """
    Returns linear system attached to the divisor D.

    EXAMPLES:
        sage: Gamma2 = graphs.CubeGraph(2)
        sage: Gamma1 = Gamma2.subgraph(vertices = ['00', '01'], edges = [('00', '01')])
        sage: f = [['00', '01', '10', '11'], ['00', '01', '00', '01']]
        sage: is_harmonic_graph_morphism(Gamma1, Gamma2, f)
        True
        sage: PhiV = matrix_of_graph_morphism_vertices(Gamma1, Gamma2, f); PhiV
        [1 0 1 0]
        [0 1 0 1]
        sage: D = vector([1,0,0,1])
        sage: PhiV*D
        (1, 1)
        sage: linear_system(PhiV*D, Gamma1)
        [(2, 0), (1, 1), (0, 2)]
        sage: linear_system(D, Gamma2)
        [(0, 2, 0, 0), (0, 0, 2, 0), (1, 0, 0, 1)]
        sage: [PhiV*x for x in linear_system(D, Gamma2)]
        [(0, 2), (2, 0), (1, 1)]

    """
    Q = Gamma.laplacian_matrix()
    CS = Q.column_space()
    N = len(D.list())
    d = sum(D.list())
    #print d
    lin_sys = []
    if d < 0:
        return lin_sys
    if (d == 0) and (D in CS):
        lin_sys = [CS(0)]
        return lin_sys
    elif (d == 0):
        return lin_sys
    S = IntegerModRing(d+1)^N
    V = QQ^N
    for v in S:
        v = V(v)
        #print D-v,v,D
        if D-v in CS:
            lin_sys.append(v)
    return lin_sys