Rank of Hadamard powers of Euclidean distance matrices

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.