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Topic: importance of eigenvalues in graph invariants
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Posts: 1
Registered: 10/21/09
importance of eigenvalues in graph invariants
Posted: Oct 21, 2009 11:43 AM
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I went through few linear algebra books and I have few
questions about eigenvalues.

1. The characteristic polynomial det|A-eI| = 0
can be solved to get eigenvalues e1, e2....

I understand that det|A-eI| has to be 0 so that
the matrix 'A-eI' is not invertible.
But looking at this equation geometrically, it means
that volume of structure formed by column vectors of
'A-eI' has to be zero. Can anyone explain how the
volume enclosed by column vectors and eigenvalues
are related to each other.

2. Can any one explain eigenvalues in terms of graph
structure. For example : We can say that determinant
is determined by number and even/odd cycles in
graphs. Is there any such relation between eigen
values and graph.


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