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for g in graphs.nauty_geng("8 -c"):

h=g.laplacian_matrix().eigenvalues()

h.sort()

show(h)

g.show()

Using this code I have generate all connected graphs on 8 vertices. Now among all these collection I need only those bicyclic graph or graphs (if it is more than one) ( A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n + 1, here we need those graphs having exactly 9 edges) on 9 edges, whose algebraic connectivity (i.e the second smallest laplacian eigenvalue) is maximum among all other bicyclic graphs on 8 vertices and 9 edges.

for g in graphs.nauty_geng("8 -c"):

if g.size()==9:

g.show()

h=g.laplacian_matrix().eigenvalues()

h.sort()

show(h)

g.show()

Using this code I have generate all connected graphs on 8 vertices. vertices and 9 edges. Now among all these collection I need only those bicyclic graph or graphs (if it is more than one) ( A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n + 1, here we need those graphs having exactly 9 edges) on 9 edges, whose algebraic connectivity (i.e the second smallest laplacian eigenvalue) is maximum among all other bicyclic graphs on 8 vertices and 9 edges.

for g in graphs.nauty_geng("8 -c"):

if g.size()==9:

g.show()

h=g.laplacian_matrix().eigenvalues()

h.sort()

show(h)

g.show()

Using this code I have generate all connected graphs on 8 vertices and 9 edges. Now among all these collection I need only those graph or graphs (if it is more than one) on 9 edges, whose algebraic connectivity (i.e the second smallest laplacian eigenvalue) is maximum among all other bicyclic graphs on 8 vertices and 9 edges.edges. Here the maximum algebraic connectivity attained by the class is 1. Now how to find those graphs [having algebraic connectivity=1] from the entire collection?

for g in graphs.nauty_geng("8 -c"):

if g.size()==9:

g.show()

h=g.laplacian_matrix().eigenvalues()

h.sort()

show(h)

g.show()

Using this The following code I have generate runs through all connected graphs on 8 vertices and 9 edges. edges, and shows each graph as well as the sorted list of its laplacian eigenvalues.

for g in graphs.nauty_geng("8 -c"):
 Now among all these collection I need    if g.size() == 9:
    g.show()
    h = g.laplacian_matrix().eigenvalues()
    h.sort()
    show(h)

How could one list only those graph or the graphs (if it is more than one) on 9 edges, whose algebraic connectivity connectivity (i.e the second smallest laplacian eigenvalue) is maximum among all other all bicyclic graphs on 8 vertices and 9 edges. Here the maximum maximum algebraic connectivity attained by the class is 1. Now how to find those graphs [having algebraic connectivity=1] from the entire collection?1.

A connected graph on $n \ge 1$ vertices is cyclic (or unicyclic) if it has $n$ edges, bicyclic if it has $n + 1$ edges.

The following code runs through all connected graphs on 8 vertices and with 9 edges, and shows each graph as well as the sorted list of its laplacian eigenvalues.

for g in graphs.nauty_geng("8 -c"):
    if g.size() == 9:
     g.show()
     h = g.laplacian_matrix().eigenvalues()
     h.sort()
     show(h)

How could one get the list only the of graphs whose algebraic connectivity (i.e (i.e. second smallest laplacian eigenvalue) is the maximum among all bicyclic graphs on 8 vertices and 9 edges. Here the maximum algebraic connectivity attained by the class is 1.among those (the maximum here being $1$)?