Editing Distance matrix (section)

== Chemistry ==
The distance matrix is a mathematical object widely used in both graphical-theoretical (topological) and geometric (topographic) versions of chemistry.<ref name=":2">{{Cite journal |last=Mihalic |first=Zlatko |date=1992 |title=The distance matrix in chemistry |journal=Journal of Mathematical Chemistry |volume=11 |pages=223–258|doi=10.1007/BF01164206 |s2cid=121181446 }}</ref> The distance matrix is used in chemistry in both explicit and implicit forms.

=== Interconversion mechanisms between two permutational isomers ===
Distance matrices were used as the main approach to depict and reveal the shortest path sequence needed to determine the rearrangement between the two permutational isomers.

=== Distance Polynomials and Distance Spectra ===
Explicit use of Distance matrices is required in order to construct the distance polynomials and distance spectra of molecular structures.

=== Structure-property model ===
Implicit use of Distance matrices was applied through the use of the distance based metric [[Wiener index|Weiner number]]/[[Wiener index|Weiner Index]] which was formulated to represent the distances in all chemical structures. The Weiner number is equal to half-sum of the elements of the distance matrix. 
[[File:WeinerNumtoDistanceMatrix.png|thumb|Conversion formula between Weiner Number and Distance Matrix|none]]

=== Graph-theoretical Distance matrix ===
Distance matrix in chemistry that are used for the 2-D realization of molecular graphs, which are used to illustrate the main foundational features of a molecule in a myriad of applications.

[[File:Chem_DistanceMtrix.png|thumb|335x335px|Labeled tree representation of C<sub>6</sub>H<sub>14</sub>'s carbon skeleton based on its distance matrix]]
# Creating a label tree that represents the [[Skeletal formula|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 [[Skeletal formula|carbon skeleton]]. The labeled tree structure of [[hexane]] (C<sub>6</sub>H<sub>14</sub>) 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 ===
[[File:Geometric_distance_matrix.png|thumb|338x338px|Geometric distance matrix for 2,4-dimethylhexane]]
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.<ref name=":2" /> The geometric-distance matrix of a molecular structure {{Math|''G''}} is a real symmetric {{Math|''n'' x ''n''}} matrix defined in the same way as a 2-D matrix. However, the matrix elements {{Math|''D''<sub>ij</sub>}} will hold a collection of shortest Cartesian distances between {{Math|''i''}} and {{Math|''j''}} in {{Math|''G''}}. 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: