How to construct the class of all connected weighted unicyclic graphs on $n$ vertices, where exactly one edge of the cycle in the unicyclic graphs

edit

The problem can be approached with the following steps:

1. Use nauty to generate all connected unicyclic graphs on $n$ vertices, like in my earlier answer. In each such graph:
   1. assign weight 1 to all edges
   2. find its unique cycle and iteratively assign weight $i$ to each of its edges
   3. use canonical labeling to eliminate duplicates

Here is a sample code:

def get_graphs(n):
  S = set()
  # iterate over all connected unicyclic graphs with n vertices
  for G in graphs.nauty_geng(options=f'-c {n} {n}:{n}'):
    # assign weight 1 to all edges of G
    for e in G.edges(labels=False):
        G.set_edge_label(e[0], e[1], 1)
    # set of edges participating in cycles
    E = set.union( *(set(c) for c in G.cycle_basis(output='edge')) )
    for e in E:
        # temporarily set weight of e to I
        G.set_edge_label(e[0], e[1], I)
        # add canonical labeling of G to set S
        S.add( G.canonical_label(edge_labels=True).copy(weighted=True,immutable=True) )
        # restore weight of e to 1
        G.set_edge_label(e[0], e[1], 1)
  return S

For example,

for H in get_graphs(4): H.show(edge_labels=True)

produces 3 graphs:

ADDED. This is how we get matrices requested in the comments (for $n=6$ as an example):

for H in get_graphs(6):
    A = H.weighted_adjacency_matrix()
    p = [(i,j) for i in range(A.nrows()) for j in range(i) if A[i,j]==I][0]
    A[p] = -I
    f = A.characteristic_polynomial()
    h = f.reverse().subs({f.variables()[0]:-f.variables()[0]})
    h /= h.leading_coefficient()
    if f==h:
        print(A,end='\n\n')

It prints the following 3 matrices:

[ 0  0  0  1  0  0]
[ 0  0  0  0  0  1]
[ 0  0  0  0  1  I]
[ 1  0  0  0  0  1]
[ 0  0  1  0  0  1]
[ 0  1 -I  1  1  0]

[ 0  0  0  0  1  0]
[ 0  0  0  1  0  0]
[ 0  0  0  0  0  1]
[ 0  1  0  0  I  1]
[ 1  0  0 -I  0  1]
[ 0  0  1  1  1  0]

[ 0  0  0  0  0  1]
[ 0  0  0  0  1  0]
[ 0  0  0  I  0  1]
[ 0  0 -I  0  0  1]
[ 0  1  0  0  0  1]
[ 1  0  1  1  1  0]