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Bipartition Polynomials, the Ising Model, and Domination in Graphs

100%
Dod M., Kotek T., Preen J., Tittmann P.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 2
335-353
EN
This paper introduces a trivariate graph polynomial that is a common generalization of the domination polynomial, the Ising polynomial, the matching polynomial, and the cut polynomial of a graph. This new graph polynomial, called the bipartition polynomial, permits a variety of interesting representations, for instance as a sum ranging over all spanning forests. As a consequence, the bipartition polynomial is a powerful tool for proving properties of other graph polynomials and graph invariants. We apply this approach to show that, analogously to the Tutte polynomial, the Ising polynomial introduced by Andrén and Markström in [3], can be represented as a sum over spanning forests.
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From the Slit-Island Method to the Ising model: Analysis of irregular grayscale objects

88%
Mazurek P., Oszutowska-Mazurek D.
International Journal of Applied Mathematics and Computer Science
|
2014
|
tom 24
|
nr 1
49-63
EN
The Slit Island Method (SIM) is a technique for the estimation of the fractal dimension of an object by determining the area-perimeter relations for successive slits. The SIM could be applied for image analysis of irregular grayscale objects and their classification using the fractal dimension. It is known that this technique is not functional in some cases. It is emphasized in this paper that for specific objects a negative or an infinite fractal dimension could be obtained. The transformation of the input image data from unipolar to bipolar gives a possibility of reformulated image analysis using the Ising model context. The polynomial approximation of the obtained area-perimeter curve allows object classification. The proposed technique is applied to the images of cervical cell nuclei (Papanicolaou smears) for the preclassification of the correct and atypical cells.
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