of applications. Creating a label tree that represents the carbon skeleton of a molecule based on its distance matrix. The distance matrix is imperative in this application because similar molecules can have a myriad of label tree variants of their carbon skeleton. The labeled tree structure of hexane (C6H14) carbon skeleton that is created based on the distance matrix in the example, has different carbon skeleton variants that affect both the distance matrix and the labeled tree Creating a labeled graph with edge weights, used in chemical graph theory, that represent molecules with hetero-atoms. Le Verrier-Fadeev-Frame (LVFF) method is a computer oriented used to speed up the process of detecting the graph center in polycyclic graphs. However, LVFF requires the input to be a diagonalized distance matrix which is easily resolved by implementing the Householder tridiagonal-QL algorithm that takes in a distance matrix and returns the diagonalized distance needed for the LVFF method. Geometric-Distance Matrix While the graph-theoretical distance matrix 2-D captures the constitutional features of the molecule, its three-dimensional (3D) character is encoded in the geometric-distance matrix. The geometric-distance matrix is a different type of distance matrix that is based on the graph-theoretical distance matrix of a molecule to represent and graph the 3-D molecule structure. The geometric-distance matrix of a molecular structure is a real symmetric matrix defined in the same way as a 2-D matrix. However, the matrix elements will hold a collection of shortest Cartesian distances between and in . Also known as topographic matrix, the geometric-distance matrix can be constructed from the known geometry of the molecule. As an example, the geometric-distance matrix of the carbon skeleton of 2,4-dimethylhexane is shown below: Other Applications Time Series Analysis Dynamic Time Warping distance matrices are utilized with the clustering and classification algorithms of a collection/group of time series objects. Examples For example, suppose these data are to be analyzed, where pixel Euclidean distance is the distance metric. The distance matrix would be: These data can then be viewed in graphic form as a heat map. In this image, black denotes a distance of 0 and white is maximal distance. See also Computer Vision Data clustering Distance set Hollow matrix Min-plus matrix multiplication References Metric geometry Bioinformatics Matrices Graph distance