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Connected non-bipartite graphs with a unique perfect matching having highest value of determinant

asked 2024-05-21 14:07:54 +0200

rewi gravatar image

updated 2024-05-21 14:08:19 +0200

Suppose we consider the class of connected non-bipartite graphs with a unique perfect matching with 10 vertices and 12 edges. Now from this collection, how one can obtain the graph(s) whose adjacency matrix has highest absolute value of its determinant?

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answered 2024-05-21 17:29:05 +0200

Max Alekseyev gravatar image

updated 2024-05-23 21:38:58 +0200

The following code tells that the max absolute value of determinant is 9 and there are 4 such graphs achieving it:

maxdet = 0
res = []
for G in graphs.nauty_geng('-c 10 12'):
    if G.is_bipartite() or abs(G.matching_polynomial()[0])!=1:
        continue
    d = abs(G.adjacency_matrix().det())
    if d>maxdet:
        res = []
        maxdet = d
    if d==maxdet:
        res.append(G)
print('Max:',maxdet)
print('#graphs:', len(res))
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Thank you..

rewi gravatar imagerewi ( 2024-05-21 18:03:43 +0200 )edit

Can you please help me in displaying those graphs, maximizing the determinant?

rewi gravatar imagerewi ( 2024-05-22 17:51:01 +0200 )edit

Add something along these lines:

for G in res:
    G.show()

You can see them at Sagecell: https://sagecell.sagemath.org/?q=czfqjg

Max Alekseyev gravatar imageMax Alekseyev ( 2024-05-22 18:28:46 +0200 )edit

ok.But I need those graphs having a unique perfect matching.. see the question

rewi gravatar imagerewi ( 2024-05-22 20:38:53 +0200 )edit

They do have a unique perfect matching. In my code this is controlled by the requirement for G.matching_polynomial()[0] to be equal -1.

Max Alekseyev gravatar imageMax Alekseyev ( 2024-05-22 22:36:21 +0200 )edit

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Asked: 2024-05-21 14:07:54 +0200

Seen: 269 times

Last updated: May 23