Eigenvalues changing at only one position [closed]

edit

for g in graphs.nauty_geng("10 -c"):
   if g.size()==11:
   A=g.laplacian_matrix().eigenvalues()
   A.sort()
   g.show()
   show(A)

I am trying to find those graphs whose laplacian eigenvalues changes only at one position, after adding an edge to the graphs in above collection of graphs given in the code.