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Graphs having highest second smallest laplacian eigen value from a collection

asked 2019-03-03 12:00:07 -0500

anonymous user


updated 2019-03-03 12:03:46 -0500

vdelecroix gravatar image

Using this code

for G in graphs(7):
    if G.girth()==4:
        L = G.laplacian_matrix().eigenvalues()

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.

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What is the definition of algebraic connectivity?

vdelecroix gravatar imagevdelecroix ( 2019-03-03 12:04:24 -0500 )edit

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answered 2019-03-04 03:48:50 -0500

rburing gravatar image

updated 2019-03-06 01:40:09 -0500

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 = algebraic_connectivity(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]:


There are 1 graphs of (maximal) algebraic connectivity 3

graph 2 with maximal a.c.

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Ok. But suppose we use this code for all connected graphs on $8$ vertices and choose girth=$5$. Then this code is not giving the correct result

Kuldeep gravatar imageKuldeep ( 2019-03-05 05:38:06 -0500 )edit

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.

rburing gravatar imagerburing ( 2019-03-05 06:58:54 -0500 )edit

@Kuldeep you're right, I got the definition of algebraic connectivity wrong because Wikipedia was not clear. I updated both Wikipedia and my answer.

rburing gravatar imagerburing ( 2019-03-06 01:45:34 -0500 )edit

No problem. But this time your answer is perfect. Very good. Thank you

Kuldeep gravatar imageKuldeep ( 2019-03-06 11:43:24 -0500 )edit

You're welcome. You can vote and accept using the buttons to the left of the answer.

rburing gravatar imagerburing ( 2019-03-07 04:46:06 -0500 )edit

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Asked: 2019-03-03 12:00:07 -0500

Seen: 82 times

Last updated: Mar 06