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Consider the class of simple connected bicyclic graphs on $10$ vertices. Let $A$ be the adjacency matrix of any such graph. Let $A(i)$ for $i\in{1,2,3,\ldots, 10}$ denote the principal submatrix of $A$, obtained by deleting the $i$-th row and $i$-th column of $A$. Now how to find those graphs from the collection for which nullity of $A(i)$ (for each $i=1,2,3,\ldots, 10$) is $1$.

Consider the class of simple connected bicyclic graphs on $10$ vertices. Let $A$ be the adjacency matrix of any such graph. Let $A(i)$ for $i\in{1,2,3,\ldots, 10}$ denote the principal submatrix of $A$, obtained by deleting the $i$-th row and $i$-th column of $A$. Now how to find those graphs from the collection for which nullity of $A(i)$ (for each $i=1,2,3,\ldots, 10$) is $1$.

def principal_submatrix(m, s, sort=False):
if sort:
    s = sorted(s)
    return m[s, s]
def principal_submatrices(m, k):
S = Subsets(range(m.ncols()), k)
return [principal_submatrix(m, s, sort=True) for s in S]
for g in graphs.nauty_geng("10 -c"):
      if g.size() == 11:
          A=g.adjacency_matrix()

Consider the class of simple connected bicyclic graphs on $10$ vertices. Let $A$ be the adjacency matrix of any such graph. Let $A(i)$ for $i\in{1,2,3,\ldots, 10}$ denote the principal submatrix of $A$, obtained by deleting the $i$-th row and $i$-th column of $A$. Now how to find those graphs from the collection for which nullity of $A(i)$ (for each $i=1,2,3,\ldots, 10$) is $1$.$1$ or equivalently rank of $A(i)$ (for each $i=1,2,3,\ldots, 10$) is $8$.

My attempt:

def principal_submatrix(m, s, sort=False):
if sort:
    s = sorted(s)
    return m[s, s]
def principal_submatrices(m, k):
S = Subsets(range(m.ncols()), k)
return [principal_submatrix(m, s, sort=True) for s in S]
for g in graphs.nauty_geng("10 -c"):
      if g.size() == 11:
          A=g.adjacency_matrix()

From this point, can any body please help me to complete the final code

Consider the class of simple connected bicyclic graphs on $10$ vertices. Let $A$ be the adjacency matrix of any such graph. Let $A(i)$ for $i\in{1,2,3,\ldots, 10}$ denote the principal submatrix of $A$, obtained by deleting the $i$-th row and $i$-th column of $A$. Now how to find those graphs from the collection for which nullity of $A(i)$ (for each $i=1,2,3,\ldots, 10$) is $1$ or equivalently rank of $A(i)$ (for each $i=1,2,3,\ldots, 10$) is $8$.

My attempt:

def principal_submatrix(m, s, sort=False):
 if sort:
     s = sorted(s)
    return m[s, s]
def principal_submatrices(m, k):
 S = Subsets(range(m.ncols()), k)
 return [principal_submatrix(m, s, sort=True) for s in S]
 for g in graphs.nauty_geng("10 -c"):
      if g.size() == 11:
          A=g.adjacency_matrix()

From this point, can any body please help me to complete the final code