In mathematical chemistry and computational biology, eigenvalues of distance matrices are also used as descriptors for determining the degree of similarity between different chemical structures or biological sequences. Since observed structures can vary in size, the spectra of corresponding distance matrices can be of different size, which makes the comparison of such structures difficult. In this paper we introduce a mathematical theory needed to support novel graphical (qualitative and visual) and numerical (quantitative and computational) representation of biological sequences. As the main result, we derive a formula for the rank of the Hadamard power of an Euclidean distance matrix.

COBISS.SI-ID: 16803929

We have constructed graphical (qualitative and visual) representations of DNA sequences as 2D maps and their numerical (quantitative and computational) analysis. The maps are obtained by transforming the four-letter sequences (where letters represent the four nucleic bases) via a spiral representation over triangular and square cells grids into a four-color map. The so constructed maps are then represented by distance matrices. We consider the use of several matrix invariants as DNA descriptors for determining the degree of similarity of a selection of DNA sequences.

COBISS.SI-ID: 16842585

In the paper we show that the bibliographic data can be transformed into a collection of compatible networks. Using network multiplication different interesting derived networks can be obtained. In defining them an appropriate normalization should be considered. The proposed approach can be applied also to other collections of compatible networks. The networks obtained from the bibliographic data bases can be large (hundreds of thousands of vertices). Fortunately they are sparse and can be still processed relatively fast. We answer the question when the multiplication of sparse networks preserves sparseness. The proposed approaches are illustrated with analyses of collection of networks on the topic "social network" obtained from the Web of Science. The works with large number of co-authors add large complete subgraphs to standard collaboration network thus bluring the collaboration structure. We show that using an appropriate normalization their effect can be neutralized. Among other, we propose a measure of collaborativness of authors with respect to a given bibliography and show how to compute the network of citations between authors and identify citation communities.

COBISS.SI-ID: 16739929