Using this code
for G in graphs(7):
if G.girth()==4:
L = G.laplacian_matrix().eigenvalues()
L.sort()
show(L)
G.show()
I have generated all graphs on $7$ vertices having girth=4. Now, from this code can we get the only unique graph having largest algebraic connectivity among all others.
For convenience we can define
algebraic_connectivity = lambda g: sorted(g.laplacian_matrix().eigenvalues())[1]
Then we can obtain at once
G = max((g for g in graphs(7) if g.girth() == 4), key=algebraic_connectivity)
But this assumes there is only one maximum. If we want to be more on the safe side:
my_graphs = [g for g in graphs(7) if g.girth() == 4]
from collections import defaultdict
acs = defaultdict(list)
for g in my_graphs:
ac = QQbar(algebraic_connectivity(g)) # ensure keys are always in QQbar, even if rational
acs[ac].append(g)
max_ac = max(acs.keys())
print('There are {} graphs of (maximal) algebraic connectivity {}'.format(len(acs[max_ac]), max_ac))
for g in acs[max_ac]:
show(g)
Output:
There are 1 graphs of (maximal) algebraic connectivity 3
Why do you say it's not correct? When I run G = max((g for g in graphs(8) if g.is_connected() and g.girth() == 5), key=algebraic_connectivity) (or more efficiently G = max((g for g in graphs.nauty_geng('8 -c') if g.girth() == 5), key=algebraic_connectivity)) (or the longer code with analogous modification) then it seems correct.
Asked: 2019-03-03 19:00:07 +0100
Seen: 558 times
Last updated: Jun 25 '20
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What is the definition of algebraic connectivity?