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Description: |
Given a graph G on n vertices the generalized
inverse eigenvalue problem for G consists in
describing the all the possible sequences of
2n-1 real numbers such that there exists a real
matrix symmetric matrix A whose graph is G, whose
eigenvalues are the first n elements of the
sequence and such that the the last n-1 numbers
of the sequence are the eigenvalues of A(v)
(the submatrix obtained from by deleting the row and column v).
The interlacing inequalities between the last n-1
numbers of a given sequence and the first n
numbers provide a necesasry condition for the existence of such matrix.
Using the implicit function theorem we will show
that strict interlacing inequalities are
sufficient for the existence of such a matrix.
These strict interlacing are necessary just for
one graph: the path (M. Fiedler, 1969)
Area(s):
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