[Joelib-devel] New JOELib2 release: bug fix + characteristic polynomials

[Joerg W. <we...@in...>]

Hi all,

minor bug fix release (atom label caching indexing for last atom in a
molecule, for some properties only) and
minor feature enhancement (characteristic polynomial).

After looking for a reasonable interpretation for eigenvalues of the
adjaceny matrix i have found the analogy to the hueckel matrix.This is
fantastic, but not really general for all weighted adjaceny matrices, with
user defined atom and bond labels.

Nontheless there exists still a much better analytical interpretation of
the (weighted) adjaceny matrix (with atom and bond labels) which is called
the characteristic polynomial.
If the polynomials (their coefficients) of two graphs are identical then
those graphs are isospectral (spectral theory of graphs).

Using the Heilbronner theorem, we can check and interpret substructural
fragments, at least on a linear fragment level.

Heilbronner theorem:
P(G)=P(G-eij)-P(G-vi-vj)-2*sum(P(G-Z))

P(G) is the characteristic polynomial in its general form
any edge eij
G-eij: Graph without this edge
G-vi-vj: Graph without edge atoms
sum(P(G-Z)): Sum over ALL rings Z, which contains eij. Please note that
this means really all rings and not only the SSSR ring set, so this is
quite nice, but hard to be calculated.

The characteristic polynomial is calculated by using the Le
Verrier-Faddeev-Frame method, which uses a framework of recursive matrix
operations.

Kind regards, Joerg Kurt Wegner

Dipl. Chem. Joerg K. Wegner
Center of Bioinformatics Tuebingen (ZBIT)
Department of Computer Architecture
Univ. Tuebingen, Sand 1, D-72076 Tuebingen, Germany
Phone: (+49/0) 7071 29 78970
Fax: (+49/0) 7071 29 5091
E-Mail: mailto:we...@in...
WWW:    http://www-ra.informatik.uni-tuebingen.de
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