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Post Undeleted by joro
Tried to fix it
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joro
joro
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I think this is false and there are many counterexamples.

Here is a sage session.

sage: ed,e=([(0, 1), (0, 2), (0, 3), (0, 4), (10, 25), (1, 3), (1, 4), (2, 35), (2, 4), (3, 45)], (20, 42))
sage: G=Graph(ed);F=Graph(G);F.delete_edge(e)
sage: Geg=G.adjacency_matrix().eigenvalues();eg.sort(reverse=1);eg
[4[2.414213562373095?, -1.732050807568878?, -0.4142135623730951?, -1, -1, -1]1.732050807568878?]
sage: Fef=F.adjacency_matrix().eigenvalues();ef.sort(reverse=1);ef
[0[2.228328215946636?, -1.360409337131395?, 0.1858853464264596?, -1, -1.645751311064591?, 3-1.645751311064591774622899504490?]

I think this is false and there are many counterexamples.

Here is a sage session.

sage: ed,e=([(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)], (2, 4))
sage: G=Graph(ed);F=Graph(G);F.delete_edge(e)
sage: G.adjacency_matrix().eigenvalues()
[4, -1, -1, -1, -1]
sage: F.adjacency_matrix().eigenvalues()
[0, -1, -1, -1.645751311064591?, 3.645751311064591?]

I think this is false and there are counterexamples.

Here is a sage session.

sage: ed,e=([(0, 2), (0, 4), (0, 5), (1, 3), (1, 5), (2, 4), (3, 5)], (0, 2))
sage: G=Graph(ed);F=Graph(G);F.delete_edge(e)
sage: eg=G.adjacency_matrix().eigenvalues();eg.sort(reverse=1);eg
[2.414213562373095?, 1.732050807568878?, -0.4142135623730951?, -1, -1, -1.732050807568878?]
sage: ef=F.adjacency_matrix().eigenvalues();ef.sort(reverse=1);ef
[2.228328215946636?, 1.360409337131395?, 0.1858853464264596?, -1, -1, -1.774622899504490?]
Post Deleted by joro
Source Link
joro
joro
  • 25.4k
  • 10
  • 66
  • 121

I think this is false and there are many counterexamples.

Here is a sage session.

sage: ed,e=([(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)], (2, 4))
sage: G=Graph(ed);F=Graph(G);F.delete_edge(e)
sage: G.adjacency_matrix().eigenvalues()
[4, -1, -1, -1, -1]
sage: F.adjacency_matrix().eigenvalues()
[0, -1, -1, -1.645751311064591?, 3.645751311064591?]